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JEE Mains | PYQ

Let \( \lambda \in \mathbb{Z} ,\ \vec{a}=\lambda\hat i +\hat j -\hat  k \) and \( \vec{b}=3\hat i -\hat j + 2\hat k \). Let \( \vec{c} \) be a vector such that \( (\vec{a}+\vec{b}+\vec{c})\times\vec{c}=\vec0, \ \vec{a}\cdot\vec{c}=-17 \) and \( \vec{b}\cdot\vec{c}=-20 \). Then \( \left| \vec{c}\times(\lambda\hat i +\hat j +\hat k) \right|^{2} \) is equal to:

If the lines \( \vec{r}=(\hat{i}-\hat{j}+\hat{k})+\lambda(3\hat{j}-\hat{k}) \) and \( \vec{r}=(\alpha\hat{i}-\hat{j})+\mu(2\hat{i}-3\hat{k}) \) are co-planar, then the distance of the plane containing these two lines from the point \( (\alpha,0,0) \) is:

Consider three vectors \( \vec{a}, \vec{b}, \vec{c} \). Let \( |\vec{a}| = 2|\vec{b}| = 3 \) and \( \vec{a} = \vec{b}\times\vec{c} \). If \( \alpha \in \left[0,\dfrac{\pi}{3}\right] \) is the angle between the vectors \( \vec{b} \) and \( \vec{c} \), then the minimum value of \( 27\,|\vec{c}-\vec{a}|^{2} \) is equal to:

Let $\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$. Let a vector $\vec{v}$ be in the plane containing $\vec{a}$ and $\vec{b}$. If $\vec{v}$ is perpendicular to the vector $3\hat{i} + 2\hat{j} - \hat{k}$ and its projection on $\vec{a}$ is $19$ units, then $|2\vec{v}|^2$ is equal to:

Let three vectors \( \vec{a}=\alpha \hat{i}+4\hat{j}+2\hat{k} \), \( \vec{b}=5\hat{i}+3\hat{j}+4\hat{k} \), \( \vec{c}=x\hat{i}+y\hat{j}+z\hat{k} \) form a triangle such that \( \vec{c}=\vec{a}-\vec{b} \) and the area of the triangle is \( 5\sqrt{6} \). If \( \alpha \) is a positive real number, then \( \left| \vec{c} \right|^{2} \) is

Let \( \hat{u} \) and \( \hat{v} \) be unit vectors inclined at an acute angle such that \( \left|\hat{u}\times\hat{v}\right|=\frac{\sqrt{3}}{2} \). If \( \vec{A}=\lambda \hat{u}+\hat{v}+(\hat{u}\times\hat{v}) \), then \( \lambda \) is equal to:

If \( (2,3,9), (5,2,1), (1,\lambda,8) \) and \( (\lambda,2,3) \) are coplanar, then the product of all possible values of \( \lambda \) is:

A vector \( \vec{a} \) has components \( 3p \) and \( 1 \) with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to new system, \( \vec{a} \) has components \( p + 1 \) and \( \sqrt{10} \), then a value of \( p \) is equal to:

If two vectors \( \vec{A} \) and \( \vec{B} \) having equal magnitude \( R \) are inclined at an angle \( \theta \), then:

Let \( \vec{a},\vec{b} \) and \( \vec{c} \) be three vectors such that \( \vec{a}=\vec{b}\times(\vec{b}\times\vec{c}) \). If magnitudes of the vectors \( \vec{a},\vec{b} \) and \( \vec{c} \) are \( \sqrt{2},1 \) and \( 2 \) respectively and the angle between \( \vec{b} \) and \( \vec{c} \) is \( \theta \) \( (0<\theta<\dfrac{\pi}{2}) \), then the value of \( 1+\tan\theta \) is equal to :

Let a unit vector \( \hat{u} = x \hat{i} + y \hat{j} + z \hat{k} \) make angles \( \dfrac{\pi}{2} \), \( \dfrac{\pi}{3} \) and \( \dfrac{2\pi}{3} \) with the vectors \( \dfrac{1}{\sqrt{2}} \hat{i} + \dfrac{1}{\sqrt{2}} \hat{k} \), \( \dfrac{1}{\sqrt{2}} \hat{j} + \dfrac{1}{\sqrt{2}} \hat{k} \) and \( \dfrac{1}{\sqrt{2}} \hat{i} + \dfrac{1}{\sqrt{2}} \hat{j} \) respectively. If \( \hat{v} = \dfrac{1}{\sqrt{2}} \hat{i} + \dfrac{1}{\sqrt{2}} \hat{k} + \dfrac{1}{\sqrt{2}} \hat{j} \), then \( \big| \hat{u} - \hat{v} \Big|^{2} \) is equal to

Two adjacent sides of a parallelogram \( PQRS \) are given by \( \overrightarrow{PQ}=\hat{j}+\hat{k} \) and \( \overrightarrow{PS}=\hat{i}-\hat{j} \). If the side \( PS \) is rotated about the point \( P \) by an acute angle \( \alpha \) in the plane of the parallelogram so that it becomes perpendicular to the side \( PQ \), then \( \sin^{2}\!\left(\dfrac{5\alpha}{2}\right)-\sin^{2}\!\left(\dfrac{\alpha}{2}\right) \) is equal to:

If the projection of the vector \( \hat{i}+2\hat{j}+\hat{k} \) on the sum of the two vectors \( 2\hat{i}+4\hat{j}-5\hat{k} \) and \( -\lambda \hat{i}+2\hat{j}+3\hat{k} \) is \( 1 \), then \( \lambda \) is equal to ________.

Let the vectors \( \vec{a}=-\hat{i}+\hat{j}+3\hat{k} \) and \( \vec{b}=\hat{i}+3\hat{j}+\hat{k} \). For some \( \lambda,\mu\in \mathbb{R} \), let \( \vec{c}=\lambda \vec{a}+\mu \vec{b} \). If \( \vec{c}\cdot(3\hat{i}-6\hat{j}+2\hat{k})=10 \) and \( \vec{c}\cdot(\hat{i}+\hat{j}+\hat{k})=-2 \), then \( |\vec{c}|^2 \) is equal to:

Let \( A, B, C \) be three points in \( xy \)-plane, whose position vector are by \( \sqrt{3}\,\hat{i}+\hat{j} \),\( \hat{i}+\sqrt{3}\,\hat{j} \) and \( a\hat{i}+(1-a)\hat{j} \) respectively with respect to the origin \( O \). If the distance of the point \( C \) from the line bisecting the angle between the vectors \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \) is \(\displaystyle \frac{9}{\sqrt{2}} \), then the sum of all the possible values of \( a \) is:

$\vec{a} = \alpha\hat{i} + \beta\hat{j} + 3\hat{k}$ , $\vec{b} = -\beta\hat{i} - \alpha\hat{j} - \hat{k}$ and $\vec{c} = \hat{i} - 2\hat{j} - \hat{k}$ such that $\vec{a} \cdot \vec{b} = 1$ and $\vec{b} \cdot \vec{c} = -3$ , then $\frac{1}{3} ((\vec{a} \times \vec{b}) \cdot \vec{c})$ is equal to _____________.

Let \( P \) be the image of the point \( Q(7,-2,5) \) in the line \( L:\; \dfrac{x-1}{2} = \dfrac{y+1}{3} = \dfrac{z}{4} \) and \( R(5,p,q) \) be a point on \( L \). Then the square of the area of \( \triangle PQR \) is \( \underline{\phantom{xxxx}} \).

Let the angle $\theta, 0 < \theta < \frac{\pi}{2}$ between two unit vectors $\hat{a}$ and $\hat{b}$ be $\sin^{-1}\left(\frac{\sqrt{65}}{9}\right)$. If the vector $\vec{c} = 3\hat{a} + 6\hat{b} + 9\left(\hat{a} \times \hat{b}\right)$, then the value of $9(\vec{c}.\hat{a}) - 3(\vec{c}.\hat{b})$ is:

Let \( \vec{a}, \vec{b}, \vec{c} \) be three coplanar concurrent vectors such that angles between any two of them is same. If the product of their magnitudes is \( 14 \) and \( (\vec{a} \times \vec{b}) \cdot (\vec{b} \times \vec{c}) + (\vec{b} \times \vec{c}) \cdot (\vec{c} \times \vec{a}) + (\vec{c} \times \vec{a}) \cdot (\vec{a} \times \vec{b}) = 168 \), then \( |\vec{a}|+|\vec{b}|+|\vec{c}| \) is equal

Let \( O \) be the origin. Let \( \overrightarrow{OP} = x \hat{i} + y \hat{j} - \hat{k} \) and \( \overrightarrow{OQ} = -\hat{i} + 2 \hat{j} + 3x \hat{k} \), \( x, y \in \mathbb{R}, \, x > 0 \), be such that \( |\overrightarrow {PQ}| = \sqrt{20} \) and the vector \( \overrightarrow{OP} \) is perpendicular to \( \overrightarrow{OQ} \). If \( \overrightarrow{OQ} = 3 \hat{i} + z \hat{j} - 7 \hat{k} \), \( z \in \mathbb{R} \), is coplanar with \( \overrightarrow{OP} \) and \( \overrightarrow{OQ} \), then the value of \( x^{2} + y^{2} + z^{2} \) is equal to:

Let \( \vec{a}=3\hat{i}+\hat{j} \) and \( \vec{b}=\hat{i}+2\hat{j}+\hat{k} \). Let \( \vec{c} \) be a vector satisfying \( \vec{a}\times (\vec{b}\times \vec{c})=\vec{b}+\lambda \vec{c} \). If \( \vec{b} \) and \( \vec{c} \) are non-parallel, then the value of \( \lambda \) is:

The sum of all values of \( \alpha \), for which the points whose position vectors are \( \hat{i}-2\hat{j}+3\hat{k} \), \( 2\hat{i}-3\hat{j}+4\hat{k} \), \( (\alpha+1)\hat{i}+2\hat{k} \) and \( 9\hat{i}+(\alpha-8)\hat{j}+6\hat{k} \) are coplanar, is equal to:

Let the vectors \( \vec{a}=(1+t)\hat{i}+(1-t)\hat{j}+\hat{k}, \ \vec{b}=(1-t)\hat{i}+(1+t)\hat{j}+2\hat{k} \) and \( \vec{c}=t\hat{i}-t\hat{j}+\hat{k}, \ t\in \mathbb{R} \) be such that for \( \alpha,\beta,\gamma \in \mathbb{R}, \ \alpha\vec{a}+\beta\vec{b}+\gamma\vec{c}=\vec{0} \Rightarrow \alpha=\beta=\gamma=0 \). Then, the set of all values of \( t \) is:

Let \( \vec{a},\vec{b},\vec{c} \) be three vectors such that \( |\vec{a}|=\sqrt{31},\;4\;|\vec{b}|=|\vec{c}|=2 \) and \( 2(\vec{a}\times\vec{b})=3(\vec{c}\times\vec{a}) \). If the angle between \( \vec{b} \) and \( \vec{c} \) is \( \dfrac{2\pi}{3} \), then \( \left(\dfrac{\vec{a}\times\vec{c}}{\vec{a}\cdot\vec{b}}\right)^{2} \) is equal to ______.

Let $\vec{a} = \hat{i} + 2\hat{j} + \lambda\hat{k}, \vec{b} = 3\hat{i} - 5\hat{j} - \lambda\hat{k}, \vec{a} \cdot \vec{c} = 7, 2\vec{b} \cdot \vec{c} + 43 = 0, \vec{a} \times \vec{c} = \vec{b} \times \vec{c}$. Then $|\vec{a} \cdot \vec{b}|$ is equal to:

Let \( \vec{a} \) and \( \vec{b} \) be the vectors along the diagonals of a parallelogram having area \( 2\sqrt{2} \). Let the angle between \( \vec{a} \) and \( \vec{b} \) be acute, \( |\vec{a}| = 1 \), and \( |\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}| \). If \( \vec{c} = 2\sqrt{2}\,(\vec{a} \times \vec{b}) - 2\vec{b} \), then an angle between \( \vec{b} \) and \( \vec{c} \) is

Let \( \vec{p}=2\hat{i}+3\hat{j}+\hat{k} \) and \( \vec{q}=\hat{i}+2\hat{j}+\hat{k} \) be two vectors. If a vector \( \vec{r}=(\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}) \) is perpendicular to each of the vectors \( (\vec{p}+\vec{q}) \) and \( (\vec{p}-\vec{q}) \) and \( |\vec{r}|=\sqrt{3} \), then \( |\alpha|+|\beta|+|\gamma| \) is equal to __________ .

If \(|\vec{a}| = 2\), \(|\vec{b}| = 5\) and \(|\vec{a} \times \vec{b}| = 8\), then \(|\vec{a} \cdot \vec{b}|\) is equal to:

Let \( \vec{a}=3\hat{i}+\hat{j}-2\hat{k},\; \vec{b}=4\hat{i}+\hat{j}+7\hat{k} \) and \( \vec{c}=\hat{i}-3\hat{j}+4\hat{k} \) be three vectors. If a vectors \( \vec{p} \) satisfies \( \vec{p}\times\vec{b}=\vec{c}\times\vec{b} \) and \( \vec{p}\cdot \vec{a}=0 \), then \( \vec{p}\cdot(\hat{i}-\hat{j}-\hat{k}) \) is equal to:

Let \( \vec{u} = \hat{i} - \hat{j} - 2\hat{k} \), \( \vec{v} = 2\hat{i} + \hat{j} - \hat{k} \), \( \vec{v} \cdot \vec{w} = 2 \) and \( \vec{v} \times \vec{w} = \vec{u} + \lambda \vec{v} \). Then \( \vec{u} \cdot \vec{w} \) is equal to:

Let \( \lambda \in \mathbb{R},\ \vec{a}=\lambda \hat{i}+2\hat{j}-3\hat{k},\ \vec{b}=\hat{i}-\lambda \hat{j}+2\hat{k} \). If \( \big((\vec{a}+\vec{b})\times(\vec{a}\times\vec{b})\big)\times(\vec{a}-\vec{b})=8\hat{i}-40\hat{j}-24\hat{k} \), then \( \left|\lambda\big(\vec{a}+\vec{b})\times(\vec{a}-\vec{b}\big)\right|^{2} \) is equal to :

Let \( \vec{a} = 2\hat{i} + 3\hat{j} + 4\hat{k} \), \( \vec{b} = \hat{i} - 2\hat{j} - 2\hat{k} \) and \( \vec{c} = -\hat{i} + 4\hat{j} + 3\hat{k} \). If \( \vec{d} \) is a vector perpendicular to both \( \vec{b} \) and \( \vec{c} \), and \( \vec{a} \cdot \vec{d} = 18 \), then \( \left|\vec{a} \times \vec{d}\right|^{2} \) is equal to:

Let \( \vec{a} \) and \( \vec{b} \) be two vectors, let \( \lvert \vec{a} \rvert =1 \), \( \lvert \vec{b} \rvert =4 \) and \( \vec{a}\cdot \vec{b}=2 \). If \( \vec{c}=(2\vec{a}\times \vec{b})-3\vec{b} \), then the value of \( \vec{b}\cdot \vec{c} \) is:

Let \( \vec{a} = 2\hat{i} + \alpha \hat{j} + \hat{k}, \ \vec{b} = -\hat{i} + \hat{k}, \ \vec{c} = \beta \hat{j} - \hat{k} \), where \( \alpha \) and \( \beta \) are integers and \( \alpha \beta = -6 \). Let the values of the ordered pair \( (\alpha , \beta) \), for which the area of the parallelogram of diagonals \( \vec{a} + \vec{b} \) and \( \vec{b} + \vec{c} \) is \( \frac{\sqrt{21}}{2} \), be \( (\alpha_{1}, \beta_{1}) \) and \( (\alpha_{2}, \beta_{2}) \). Then \( \alpha_{1}^{2} + \beta_{1}^{2} - \alpha_{2}\beta_{2} \) is equal to

Let \( \vec{a} \) and \( \vec{b} \) be two vectors such that \( \Big| \vec{b} \Big| = 1 \) and \( \Big| \vec{b} \times \vec{a}\Big| = 2 \). Then \( \Big|( \vec{b} \times \vec{a}) - \vec{b} \Big|^{2} \) is equal to:

The square of the distance of the point of intersection of the lines \( \vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(a\hat{i} - \hat{j}),\, a \ne 0 \) and \( \vec{r} = (4\hat{i} - \hat{k}) + \mu(2\hat{i} + a\hat{k}) \) from the origin is:

Let \( \hat{a} \) and \( \hat{b} \) be two unit vectors such that \( \left|(\hat{a}+\hat{b})+2(\hat{a}\times \hat{b})\right|=2 \). If \( \theta \in (0,\pi) \) is the angle between \( \hat{a} \) and \( \hat{b} \), then among the statements:
(S1): \( 2\left|\hat{a}\times \hat{b}\right|=\left|\hat{a}-\hat{b}\right| \)
(S2): The projection of \( \hat{a} \) on \( (\hat{a}+\hat{b}) \) is \( \frac{1}{2} \)

Let \( \vec{a}=\alpha \hat{i}+\hat{j}-\hat{k} \) and \( \vec{b}=2\hat{i}+\hat{j}-\alpha \hat{k}, \, \alpha>0 \). If the projection of \( \vec{a}\times\vec{b} \) on the vector \( -\hat{i}+2\hat{j}-2\hat{k} \) is \( 30 \), then \( \alpha \) is equal to:

If the four points, whose position vectors are \( 3\hat{i}-4\hat{j}+2\hat{k} \), \( \hat{i}+2\hat{j}-\hat{k} \), \( -2\hat{i}-\hat{j}+3\hat{k} \) and \( 5\hat{i}-2\alpha \hat{j}+4\hat{k} \) are coplanar, then \( \alpha \) is equal to:

Let \( A, B, C \) be three points whose position vectors respectively are
\( \vec{a}=\hat{i}+4\hat{j}+3\hat{k} \)
\( \vec{b}=2\hat{i}+\alpha \hat{j}+4\hat{k}, \ \alpha \in \mathbb{R} \)
\( \vec{c}=3\hat{i}-2\hat{j}+5\hat{k} \)
If \( \alpha \) is the smallest positive integer for which \( \vec{a}, \vec{b}, \vec{c} \) are non-collinear, then the length of the median, in \( \Delta ABC \), through \( A \) is:

Let \( \hat{a},\ \hat{b} \) be unit vectors. If \( \vec{c} \) be a vector such that the angle between \( \hat{a} \) and \( \vec{c} \) is \( \frac{\pi}{12} \), and \( \hat{b} = \vec{c} + 2(\vec{c} \times \hat{a}) \), then \( \lvert 6\,\vec{c} \rvert^{2} \) is equal to:

Let \( \vec{a} = 2\hat{i} + \hat{j} - \hat{k}, \ \vec{b} = \big((\vec{a} \times (\hat{i} + \hat{j})) \times \hat{i}\big) \times \hat{i} \). Then the square of the projection of \( \vec{a} \) on \( \vec{b} \) is:

Let \( \vec{a} = 2\hat{i} - \hat{j} + 5\hat{k} \) and \( \vec{b} = \alpha \hat{i} + \beta \hat{j} + 2\hat{k} \). If \( \left((\vec{a} \times \vec{b}) \times \hat{i}\right) \cdot \hat{k} = \dfrac{23}{2} \), then \( \left| \vec{b} \times 2\hat{i} \right| \) is equal to:

If \( A(3,1,-1), B\!\left(\frac{5}{3},\frac{7}{3},\frac{1}{3}\right) \), \( C(2,2,1) \) and \( D\!\left(\frac{10}{3},\frac{2}{3},-\frac{1}{3}\right) \) are the vertices of a quadrilateral \( ABCD \), then its area is:

Let \( \vec{a} = -5\hat{i} + \hat{j} - 3\hat{k} \), \( \vec{b} = \hat{i} + 2\hat{j} - 4\hat{k} \) and \( \vec{x} = (((\vec{a} \times \vec{b}) \times \hat{i}) \times \hat{i}) \). Then \( \vec{c} \cdot (-\hat{i} + \hat{j} + \hat{k}) \) is equal to:

Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k} , \\ \vec{b} = 2\hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{d} = \vec{a} \times \vec{b} \). If \( \vec{c} \) is a vector such that \( \vec{a}\cdot\vec{c} = |\vec{c}| \), \( |\vec{c} - 2\vec{a}|^{2} = 8 \) and the angle between \( \vec{d} \) and \( \vec{c} \) is \( \frac{\pi}{4} \), then \( |10 - 3\vec{b}\cdot\vec{c}| + |\vec{d}\times\vec{c}|^{2} \) is equal to ______.

If four distinct points with position vectors \( \vec{a}, \vec{b}, \vec{c} \) and \( \vec{d} \) are coplanar, then \( [\vec{a}\ \vec{b}\ \vec{c}] \) is equal to:

For three unit vectors \( \vec{a}, \vec{b}, \vec{c} \) satisfying \( \lvert \vec{a}-\vec{b} \rvert^{2} + \lvert \vec{b}-\vec{c} \rvert^{2} + \lvert \vec{c}-\vec{a} \rvert^{2} = 9 \) and \( \lvert 2\vec{a} + k\vec{b} + k\vec{c} \rvert = 3 \), the positive value of \( k \) is:

If \( \vec{a}=\hat{i}+2\hat{k},\,\vec{b}=\hat{i}+\hat{j}+\hat{k},\,\vec{c}=7\hat{i}-3\hat{j}+4\hat{k},\,\vec{r}\times\vec{b}+\vec{b}\times\vec{c}=\vec{0} \) and \( \vec{r}\cdot\vec{a}=0 \). Then \( \vec{r}\cdot\vec{c} \) is equal to:

Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k}, \ \vec{b} = 3\hat{i} + 2\hat{j} - \hat{k}, \ \vec{c} = \lambda \hat{j} + \mu \hat{k} \) and \( \hat{d} \) be a unit vector such that \( \vec{a} \times \hat{d} = \vec{b} \times \hat{d} \) and \( \vec{c} \cdot \hat{d} = 1 \). If \( \vec{c} \) is perpendicular to \( \vec{a} \), then \( |3\lambda \hat{d} + \mu \vec{c}|^{2} \) is equal to _________.

Let \( \vec{a} = 4\hat{i} + 3\hat{j} \) and \( \vec{b} = 3\hat{i} - 4\hat{j} + 5\hat{k} \). If \( \vec{c} \) is a vector such that \( \vec{c} \cdot (\vec{a} \times \vec{b}) + 25 = 0 \), \( \vec{c} \cdot (\hat{i} + \hat{j} + \hat{k}) = 4 \), and projection of \( \vec{c} \) on \( \vec{a} \) is \( 1 \), then the projection of \( \vec{c} \) on \( \vec{b} \) equals:

Let a vector \( \vec{a} \) has magnitude \( 9 \). Let a vector \( \vec{b} \) be such that for every \( (x,y)\in \mathbb{R}\times\mathbb{R}-\{(0,0)\} \), the vector \( (x\vec{a}+y\vec{b}) \) is perpendicular to the vector \( (6y\vec{a}-18x\vec{b}) \). Then the value of \( \left|\vec{a}\times\vec{b}\right| \) is equal to:

A vector has magnitude same as that of \( \vec{A} = 3\hat{i} + 4\hat{j} \) and is parallel to \( \vec{B} = 4\hat{i} + 3\hat{j} \). The x and y components of this vector in first quadrant are \( x \) and \( 3 \) respectively were \( x = \) ______.

Let \( \vec{a}, \vec{b}, \vec{c} \) be three vectors such that \( \vec{a}\times\vec{b} = 2(\vec{a}\times\vec{c}) \). If \( |\vec{a}| = 1 \), \( |\vec{b}| = 4 \), \( |\vec{c}| = 2 \), and the angle between \( \vec{b} \) and \( \vec{c} \) is \( 60^\circ \), then \( |\vec{a}\cdot\vec{c}| \) is equal to:

Let \( ABCD \) be a quadrilateral. If \( E \) and \( F \) are the mid points of the diagonals \( AC \) and \( BD \) respectively and \( (\overrightarrow{AB}-\overrightarrow{BC})+(\overrightarrow{AD}-\overrightarrow{DC}) = k\,\overrightarrow{FE} \), then \( k \) is equal to:

Let \( \vec{a}=\alpha \hat{i}+\hat{j}+\beta \hat{k} \) and \( \vec{b}=3\hat{i}-5\hat{j}+4\hat{k} \) be two vectors, suthat \( \vec{a}\times \vec{b}=-\hat{i}+9\hat{j}+12\hat{k} \). Then the projection of \( \vec{b}-2\vec{a} \) on \( \vec{b}+\vec{a} \) is equal to:

In a triangle \( ABC \), if \( | \overrightarrow{BC} | = 8, | \overrightarrow{CA} | = 7, | \overrightarrow{AB} | = 10 \), then the projection of the vector \( \overrightarrow{AB} \) on \( \overrightarrow{AC} \) is equal to:

Let \( \vec{a}=-\hat{i}-\hat{j}+\hat{k} \), \( \vec{a}\cdot\vec{b}=1 \) and \( \vec{a}\times\vec{b}=\hat{i}-\hat{j} \). Then \( \vec{a}-6\vec{b} \) is equal to:

Let \( S \) be the set of all \( (\lambda,\mu) \) for which the vectors \( \lambda \hat{i}-\hat{j}+\hat{k} \), \( \hat{i}+2\hat{j}+\mu \hat{k} \) and \( 3\hat{i}-4\hat{j}+5\hat{k} \), where \( \lambda-\mu=5 \), are coplanar, then \( \sum\limits_{(\lambda,\mu\in S)} 80(\lambda^{2}+\mu^{2}) \) is equal to:

Let \( \vec{a} \) and \( \vec{b} \) be the vectors of the same magnitude such that \( \dfrac{|\vec{a}+\vec{b}| + |\vec{a}-\vec{b}|}{|\vec{a}+\vec{b}| - |\vec{a}-\vec{b}|} = \sqrt{2} + 1 \). Then \( \dfrac{|\vec{a}+\vec{b}|^{2}}{|\vec{a}|^{2}} \) is:

The vector \( \vec{a} = -\hat{\imath} + 2\hat{\jmath} + \hat{k} \) is rotated through a right angle, passing through the y-axis in its way and the resulting vector is \( \vec{b} \). Then the projection of \( 3\vec{a} + \sqrt{2}\,\vec{b} \) on \( \vec{c} = 5\hat{\imath} + 4\hat{\jmath} + 3\hat{k} \) is:

If \( \vec{a}\cdot\vec{b} = 1 \), \( \vec{b}\cdot\vec{c} = 2 \) and \( \vec{c}\cdot\vec{a} = 3 \), then the value of \( \left[ \vec{a}\times(\vec{b}\times\vec{c}), \; \vec{b}\times(\vec{c}\times\vec{a}), \; \vec{c}\times(\vec{b}\times\vec{a}) \right] \) is:

Let \( \vec{\alpha} = 4\hat{i} + 3\hat{j} + 5\hat{k} \) and \( \vec{\beta} = \hat{i} + 2\hat{j} - 4\hat{k} \). Let \( \vec{\beta}_1 \) be parallel to \( \vec{\alpha} \) and \( \vec{\beta}_2 \) be perpendicular to \( \vec{\alpha} \).

If \( \vec{\beta} = \vec{\beta}_1 + \vec{\beta}_2 \), then the value of \( 5\vec{\beta}_2 \cdot (\hat{i}, \hat{j}, \hat{k}) \) is:

If \( \vec{a}, \vec{b}, \vec{c} \) are three non-zero vectors and \( \hat{n} \) is a unit vector perpendicular to \( \vec{c} \) such that \( \vec{a}=\alpha \vec{b}-\hat{n} \), \( (\alpha \neq 0) \) and \( \vec{b}\cdot \vec{c}=12 \), then \( \left\lvert \vec{c}\times(\vec{a}\times\vec{b}) \right\rvert \) is equal to:

The vector equation of the plane passing through the intersection of the planes \( \vec r \cdot (\hat i+\hat j+\hat k)=1 \) and \( \vec r \cdot (\hat i-2\hat j)=-2 \), and the point \( (1,0,2) \) is:

Let \( \vec a=4\hat i-\hat j+3\hat k \), \( \vec b=10\hat i+2\hat j-\hat k \) and a vector \( \vec c \) be such that \( 2(\vec a\times \vec b)+3(\vec b\times \vec c)=\vec 0 \). If \( \vec a\cdot \vec c=15 \), then \( \vec c\cdot(\hat i+\hat j-3\hat k) \) is equal to:

Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{b} = 3\hat{i} + \hat{j} - \hat{k}$ and $\vec{c}$ be three vectors such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$. If the vector $\vec{c}$ is perpendicular to $\vec{b}$ and $\vec{a} \cdot \vec{c} = 5$, then $|\vec{c}|$ is equal to:

Let \( \vec{a} = -\hat{i} + 2\hat{j} + 2\hat{k} \), \( \vec{b} = 8\hat{i} + 7\hat{j} - 3\hat{k} \) and \( \vec{c} \) be a vector such that \( \vec{a} \times \vec{c} = \vec{b} \). If \( \vec{c} \cdot (\hat{i} + \hat{j} + \hat{k}) = 4 \), then \( \left|\vec{a} + \vec{c}\right|^2 \) is equal to:

Let \(  ABC \) be a triangle such that \( \overrightarrow{BC}=\vec{a} \), \( \overrightarrow{CA}=\vec{b} \), \( \overrightarrow{AB}=\vec{c} \), \( |\vec{a}|=6\sqrt{2} \), \( |\vec{b}|=2\sqrt{3} \) and \( \vec{b}\cdot\vec{c}=12 \). Consider the statements:
(S1): \( \big|(\vec{a}\times\vec{b})+(\vec{c}\times\vec{b})\big| - |\vec{c}| = 6(2\sqrt{2}-1) \)
(S2): \( \angle ACB = \cos^{-1}\!\left(\sqrt{\dfrac{2}{3}}\right) \)
Then

Let \( \vec{a}=\hat{i}+2\hat{j}-3\hat{k} \) and \( \vec{b}=2\hat{i}-3\hat{j}+5\hat{k} \). If \( \vec{r}\times \vec{a}=\vec{b}\times \vec{r},\ \vec{r}\cdot(\alpha \hat{i}+2\hat{j}+\hat{k})=3 \) and \( \vec{r}\cdot(2\hat{i}+5\hat{j}-\alpha \hat{k})=-1, \ \alpha \in \mathbb{R} \), then the value of \( \alpha+|\vec{r}|^{2} \) is equal to:

Let the shortest distance between the lines \( \dfrac{x-3}{3} = \dfrac{y-\alpha}{-1} = \dfrac{z-3}{1} \) and \( \dfrac{x+3}{-3} = \dfrac{y+7}{2} = \dfrac{z-\beta}{4} \) be \( 3\sqrt{30} \). Then the positive value of \( 5\alpha + \beta \) is:

A vector \( \vec{a} \) is parallel to the line of intersection of the plane determined by the vectors \( \hat{\imath}, \ \hat{\imath}+\hat{\jmath} \) and the plane determined by the vectors \( \hat{\imath}-\hat{\jmath}, \ \hat{\imath}+\hat{k} \). The obtuse angle between \( \vec{a} \) and the vector \( \vec{b}=\hat{\imath}-2\hat{\jmath}+2\hat{k} \) is:

Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k} \), \( \vec{b} = -\hat{i} - 8\hat{j} + 2\hat{k} \) and \( \vec{c} = 4\hat{i} + c_2 \hat{j} + c_3 \hat{k} \) be three vectors such that \( \vec{b} \times \vec{a} = \vec{c} \times \vec{a} \). If the angle between vector \( \vec{c} \) and the vector \( 3\hat{i} + 4\hat{j} + \hat{k} \) is \( \theta \), then the greatest integer less than or equal to \( \tan^2 \theta \) is \( \_\_\_\_\_\_\_\_\_\_ \).

Let \( \vec{a} = 2\hat{i} + 5\hat{j} - \hat{k} \), \( \vec{b} = 2\hat{i} - 2\hat{j} + 2\hat{k} \) and \( \vec{c} \) be three vectors such that \( (\vec{c} + \hat{i}) \times (\vec{a} + \vec{b} + \hat{i}) = \vec{a} \times (\vec{c} + \hat{i}) \). If \( \vec{a}\cdot\vec{c} = -29 \), then \( \vec{c}\cdot(-2\hat{i} + \hat{j} + \hat{k}) \) is equal to:

Let \( \vec{v} = \alpha \hat{i} + 2\hat{j} - 3\hat{k} , \; \vec{w} = 2\alpha \hat{i} + \hat{j} - \hat{k} \) and \( \vec{u} \) be a vector such that \( |\vec{u}| = \alpha > 0 \). If the minimum value of the scalar triple product \( [\vec{u}\, \vec{v}\, \vec{w}] \) is \( -\alpha \sqrt{3401} \), and \( \left| \vec{u}\, \hat{i} \right| = \frac{m}{n} \) where \( m \) and \( n \) are coprime natural numbers, then \( m + n \) is equal to ________.

Let \( \vec{a} = 2\hat{i} - \hat{j} + \hat{k} \) and \( \vec{b} = \lambda \hat{j} + 2\hat{k}, \lambda \in \mathbb{Z} \) be two vectors. Let \( \vec{c} = \vec{a} \times \vec{b} \) and \( \vec{d} \) be a vector of magnitude \( 2 \) in \( yz \)-plane. If \( |\vec{c}| = \sqrt{53} \), then the maximum possible value of \( (\vec{c}\cdot \vec{d})^2 \) is equal to:

Let the point \( A \) divide the line segment joining the points \( P(-1,-1,2) \) and \( Q(5,5,10) \) internally in the ratio \( r:1 \, (r>0) \). If \( O \) is the origin and \( {\overrightarrow {OQ} \cdot \overrightarrow{OA}} - \frac{1}{5} \left|\overrightarrow{ OP} \times \overrightarrow{ OA} \right|^{2} = 10 \), then the value of \( r \) is:

Let \( \vec{a} = -\vec{i} + \vec{j} + 2\vec{k},\ \vec{b} = \vec{i} - \vec{j} - 3\vec{k},\ \vec{c} = \vec{a} \times \vec{b} \) and \( \vec{d} = \vec{c} \times \vec{a}. \) Then \( (\vec{a} - \vec{b}) \cdot \vec{d} \) is equal to:

Let \( \vec{a} \) and \( \vec{b} \) be two-unit vectors such that the angle between them is \( \frac{\pi}{3} \). If \( \lambda \vec{a} + 2\vec{b} \) and \( 3\vec{a} - \lambda \vec{b} \) are perpendicular to each other, then the number of values of \( \lambda \) in \( [-1, 3] \) is:

Let \( \vec{a} = 2\hat{i} + \hat{j} + \hat{k} \), and \( \vec{b} \) and \( \vec{c} \) be two nonzero vectors such that \( |\vec{a}+\vec{b}+\vec{c}| = |\vec{a}+\vec{b}-\vec{c}| \) and \( \vec{b}\cdot\vec{c} = 0 \). Consider the following two statements:
(A) \( |\vec{a}+\lambda \vec{c}| \ge |\vec{a}| \) for all \( \lambda \in \mathbb{R} \)
(B) \( \vec{a} \) and \( \vec{c} \) are always parallel.

Let \( \vec{a}=\hat{i}+2\hat{j}+3\hat{k} \) and \( \vec{b}=\hat{i}+\hat{j}-\hat{k} \). If \( \vec{c} \) is a vector such that \( \vec{a}\cdot\vec{c}=11 \), \( \vec{b}\cdot(\vec{a}\times\vec{c})=27 \) and \( \vec{b}\cdot\vec{c}=-\sqrt{3}\,|\vec{b}| \), then \( \left|\vec{a}\times\vec{c}\right|^{2} \) is equal to ______.

Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k}, \\ \vec{b} = 2\hat{i} + 4\hat{j} - 5\hat{k} \) and \( \vec{c} = x\hat{i} + 2\hat{j} + 3\hat{k}, \\ x \in \mathbb{R}. \) If \( \vec{d} \) is the unit vector in the direction of \( \vec{b} + \vec{c} \) such that \( \vec{a} \cdot \vec{d} = 1 \), then \( (\vec{a} \times \vec{b}) \cdot \vec{c} \) is equal to:

Let for a triangle \(ABC\),
\(\overrightarrow{AB}=-2\hat{i}+\hat{j}+3\hat{k}\)
\(\overrightarrow{CB}=\alpha\hat{i}+\beta\hat{j}+\gamma\hat{k}\)
\(\overrightarrow{CA}=4\hat{i}+3\hat{j}+\delta\hat{k}\).
If \(\delta>0\) and the area of the triangle \(ABC\) is \(5\sqrt{6}\), then \(\overrightarrow{CB}\cdot\overrightarrow{CA}\) is equal to :

Let \( \vec{a} = 2\hat{i} + 7\hat{j} - \hat{k} \), \( \vec{b} = 3\hat{i} + 5\hat{k} \) and \( \vec{c} = \hat{i} - \hat{j} + 2\hat{k} \). Let \( \vec{d} \) be a vector which is perpendicular to both \( \vec{a} \) and \( \vec{b} \), and \( \vec{c} \cdot \vec{d} = 12 \). Then \( (-\hat{i} + \hat{j} - \hat{k}) \cdot (\vec{c} \times \vec{d}) \) is equal to:

Let \( \vec{a} = \hat{i} + 2\hat{j} + \hat{k}, \ \vec{b} = 3\hat{i} - 3\hat{j} + 3\hat{k}, \ \vec{c} = 2\hat{i} - \hat{j} + 2\hat{k} \) and \( \vec{d} \) be a vector such that \( \vec{b} \times \vec{d} = \vec{c} \times \vec{d} \) and \( \vec{a} \cdot \vec{d} = 4 \). Then \( \left| \left( \vec{a} \times \vec{d} \right) \right|^{2} \) is equal to __________.

Let a vector \( \vec{a} \) be coplanar with vectors \( \vec{b}=2\hat{i}+\hat{j}+\hat{k} \) and \( \vec{c}=\hat{i}-\hat{j}+\hat{k} \). If \( \vec{a} \) is perpendicular to \( \vec{d}=3\hat{i}+2\hat{j}+6\hat{k} \), and \( |\vec{a}|=\sqrt{10} \). Then a possible value of \( [\vec{a}\ \vec{b}\ \vec{c}] + [\vec{a}\ \vec{b}\ \vec{d}] + [\vec{a}\ \vec{c}\ \vec{d}] \) is equal to:

Let \( \vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k} \) and \( \vec{b} = 7\hat{i} + \hat{j} - 6\hat{k} \). If \( \vec{r} \times \vec{a} = \vec{r} \times \vec{b}, \ \vec{r} \cdot (\hat{i} + 2\hat{j} + \hat{k}) = 3 \), then \( \vec{r} \cdot (2\hat{i} - 3\hat{j} + \hat{k}) \) is equal to:

Let $\vec{a} = \alpha\hat{i} + 2\hat{j} - \hat{k}$ and $\vec{b} = -2\hat{i} + \alpha\hat{j} + \hat{k}$, where $\alpha \in \mathbb{R}$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\vec{a}$ and $\vec{b}$ is $\sqrt{15(\alpha^2+4)}$, then the value of $2|\vec{a}|^2 + (\vec{a}\cdot\vec{b})|\vec{b}|^2$ is equal to:

Let \( \vec{a} \) and \( \vec{b} \) be two vectors such that \(|\vec{a}+\vec{b}|^{2}=|\vec{a}|^{2}+2|\vec{b}|^{2}\), \( \vec{a}\cdot\vec{b}=3 \) and \( |\vec{a}\times\vec{b}|^{2}=75 \). Then \( |\vec{a}|^{2} \) is equal to __________.

Let \( \vec{c} \) and \( \vec{d} \) be vectors such that \( |\vec{c} + \vec{d}| = \sqrt{29} \) and \( \vec{c} \times (2\hat{i} + 3\hat{j} + 4\hat{k}) = (2\hat{i} + 3\hat{j} + 4\hat{k}) \times \vec{d} \). If \( \lambda_1, \lambda_2 \) \( (\lambda_1 > \lambda_2) \) are the possible values of \( (\vec{c} + \vec{d}) \cdot (-7\hat{i} + 2\hat{j} + 3\hat{k}) \), then the equation \( K^2 x^2 + (K^2 - 5K + \lambda_1)xy + \left(3K + \frac{\lambda_2}{2}\right) y^2 - 8x + 12y + \lambda_2 = 0 \) represents a circle, for \( K \) equal to:

Let \( \vec{a} = 2\hat{i} - 3\hat{j} + \hat{k} \), \( \vec{b} = 3\hat{i} + 2\hat{j} + 5\hat{k} \) and a vector \( \vec{c} \) be such that \( (\vec{a} - \vec{c}) \times \vec{b} = -18\hat{i} - 3\hat{j} + 12\hat{k} \) and \( \vec{a} \cdot \vec{c} = 3 \). If \( \vec{b} \times \vec{c} = \vec{d} \), then \( \left| \vec{a} \cdot \vec{d} \right| \) is equal to:

Let \( \vec{c} \) be the projection vector of \( \vec{b}=\lambda \hat{i}+4\hat{k},\, \lambda>0 \), on the vector \( \vec{a}=\hat{i}+2\hat{j}+2\hat{k} \). If \( |\vec{a}+\vec{c}|=7 \), then the area of the parallelogram formed by the vectors \( \vec{b} \) and \( \vec{c} \) is ______.

Let \( \vec{a}, \ \vec{b} \) and \( \vec{c} \) be three non-zero non-coplanar vectors. Let the position vectors of four points \( A, B, C \) and \( D \) be \( \vec{a} - \vec{b} + \vec{c} \), \( \lambda \vec{a} - 3\vec{b} + 4\vec{c} \), \( -\vec{a} + 2\vec{b} - 3\vec{c} \) and \( 2\vec{a} - 4\vec{b} + 6\vec{c} \) respectively. If \( \overrightarrow{AB}, \ \overrightarrow{AC} \) and \( \overrightarrow{AD} \) are coplanar, then \( \lambda \) is equal to __________ .

Let \( a,b,c \) be three distinct real numbers, none equal to one. If the vectors \( ai + j + k, \ i + bj + k \) and \( i + j + ck \) are coplanar, then \(\displaystyle \frac{1}{1-a} + \frac{1}{1-b} + \frac{1}{1-c} \) is equal to:

Let the vectors \( \vec{a}, \vec{b}, \vec{c} \) represent three coterminous edges of a parallelepiped of volume \( V \). Then the volume of the parallelepiped, whose coterminous edges are represented by \( \vec{a}, \vec{b}, \vec{c} \) and \( \vec{a} + 2\vec{b} + 3\vec{c} \) is equal to:

Let \( \vec{a}=\hat{i}+2\hat{j}+\hat{k} \), \( \vec{b}=3(\hat{i}-\hat{j}+\hat{k}) \). Let \( \vec{c} \) be the vector such that \( \vec{a}\times\vec{c}=\vec{b} \) and \( \vec{a}\cdot\vec{c}=3 \). Then \( \vec{a}\cdot\left((\vec{c}\times\vec{b})-\vec{b}-\vec{c}\right) \) is equal to:

Let \( \vec a=2\hat i-7\hat j+5\hat k \), \( \vec b=\hat i+\hat k \) and \( \vec c=\hat i+2\hat j-3\hat k \) be three given vectors. If \( \vec r \) is a vector such that \( \vec r\times \vec a=\vec c\times \vec a \) and \( \vec r\cdot \vec b=0 \), then \( |\vec r| \) is equal to:

If \( \vec{a} \) is a nonzero vector such that its projections on the vectors $2\hat{i} - \hat{j} + 2\hat{k}, \hat{i} + 2\hat{j} - 2\hat{k}$ and $\hat{k}$ are equal, then a unit vector along \( \vec{a} \) is:

For \( p>0 \), a vector \( \vec{v}_{2}=2\hat{i}+(p+1)\hat{j} \) is obtained by rotating the vector \( \vec{v}_{1}=\sqrt{3}\,p\,\hat{i}+\hat{j} \) by an angle \( \theta \) about origin in counter clockwise direction. If \( \tan\theta=\dfrac{(\alpha\sqrt{3}-2)}{(4\sqrt{3}+3)} ,\) then the value of \( \alpha \) is equal to ______.

Between the following two statements:
Statements I : Let \( \vec{a} = \hat{i} + 2\hat{j} - 3\hat{k} \) and \( \vec{b} = 2\hat{i} + \hat{j} - \hat{k} \). Then the vector \( \vec{r} \) satisfying \( \vec{a} \times \vec{r} = \vec{a} \times \vec{b} \) and \( \vec{a} \cdot \vec{r} = 0 \) is of magnitude \( \sqrt{10} \).
Statement II : In a triangle \( ABC \), \( \cos 2A + \cos 2B + \cos 2C \ge -\frac{3}{2} \).

Let \( \vec{a} \) and \( \vec{b} \) be two vectors such that \( |\vec{a}| = \sqrt{14} \), \( |\vec{b}| = \sqrt{6} \) and \( |\vec{a} \times \vec{b}| = \sqrt{48} \). Then \( (\vec{a} \cdot \vec{b})^{2} \) is equal to _____.

Let \( |\vec{a}|=2 \), \( |\vec{b}|=3 \) and the angle between the vectors \( \vec{a} \) and \( \vec{b} \) be \( \dfrac{\pi}{4} \). Then \( \left|(\vec{a}+2\vec{b})\times(2\vec{a}-3\vec{b})\right|^{2} \) is equal to:

Let \( \vec{a}=6\hat{i}+9\hat{j}+12\hat{k},\; \vec{b}=\alpha \hat{i}+11\hat{j}-2\hat{k} \) and \( \vec{c} \) be vectors such that \( \vec{a}\times\vec{c}=\vec{a}\times\vec{b} \). If \( \vec{a}\cdot\vec{c}=-12, \; \vec{c}\cdot(\hat{i}-2\hat{j}+\hat{k})=5 \), then \( \vec{c}\cdot(\hat{i}+\hat{j}+\hat{k}) \) is equal to ________.

Let $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$ be two vectors such that $|\vec{a}| = 1$, $\vec{a} \cdot \vec{b} = 2$ and $|\vec{b}| = 4$. If $\vec{c} = 2(\vec{a} \times \vec{b}) - 3\vec{b}$, then the angle between $\vec{b}$ and $\vec{c}$ is equal to:

When vector \( \vec{A} = 2\hat{i} + 3\hat{j} + 2\hat{k} \) is subtracted from vector \( \vec{B} \), it gives a vector equal to \( 2\hat{j} \). Then the magnitude of vector \( \vec{B} \) will be :

Let \( \vec{a} = 3\hat{i} + \hat{j} - \hat{k} \) and \( \vec{c} = 2\hat{i} - 3\hat{j} + 3\hat{k} \). If \( \vec{b} \) -is a vector such that \( \vec{a} = \vec{b} \times \vec{c} \) -and \( |\vec{b}|^2 = 50 \), then \( \left| 72 - |\vec{b} + \vec{c}|^2 \right| \) is equal to ______.

Let \( \vec{a}=a_{1}\hat{i}+a_{2}\hat{j}+a_{3}\hat{k},\ a_{i}>0,\ i=1,2,3 \) be a vector which makes equal angles with the coordinate axes \( OX \), \( OY \) and \( OZ \). Also, let the projection of \( \vec{a} \) on the vector \( 3\hat{i}+4\hat{j} \) be \( 7 \). Let \( \vec{b} \) be a vector obtained by rotating \( \vec{a} \) with \( 90^\circ \). If \( \vec{a},\vec{b} \) and x-axis are coplanar, then projection of vector \( \vec{b} \) on \( 3\hat{i}+4\hat{j} \) is equal to:

For \( \lambda > 0 \), let \( \theta \) be the angle between the vectors \( \vec{a} = \hat{i} + \lambda \hat{j} - 3\hat{k} \) and \( \vec{b} = 3\hat{i} - \hat{j} + 2\hat{k} \). If the vectors \( \vec{a} + \vec{b} \) and \( \vec{a} - \vec{b} \) are mutually perpendicular, then the value of \( (14\cos\theta)^2 \) is equal to:

Let \( \vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k} \), \( \vec{b} = 3\hat{i} + 4\hat{j} - 5\hat{k} \) and a vector \( \vec{c} \) be such that \( \vec{a} \times (\vec{b} \times \vec{c}) + \vec{b} \times \vec{c} = \hat{i} + 8\hat{j} + 13\hat{k} \). If \( \vec{a} \cdot \vec{c} = 13 \), then \( \left(24 - \vec{b} \cdot \vec{c}\right) \) is equal to ________.

Let \( \vec{a}, \vec{b}, \vec{c} \) be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle \( \theta \), with the vector \( \vec{a} + \vec{b} + \vec{c} \). Then \( 36\cos^2 2\theta \) is equal to ____________.

Let \( \vec{a} = \hat{i} - 2\hat{j} + 3\hat{k} \), \( \vec{b} = \hat{i} + \hat{j} + \hat{k} \) and \( \vec{c} \) be a vector such that \( \vec{a} + (\vec{b} \times \vec{c}) = \vec{0} \) and \( \vec{b} \cdot \vec{c} = 5 \). Then, the value of \( 3(\vec{c} \cdot \vec{a}) \) is equal to ______.

Let \( \vec{a} = \hat{i} + 5\hat{j} + \alpha \hat{k} \), \( \vec{b} = \hat{i} + 3\hat{j} + \beta \hat{k} \) and \( \vec{c} = -\hat{i} + 2\hat{j} - 3\hat{k} \) be three vectors such that, \( \lvert \vec{b} \times \vec{c} \rvert = 5\sqrt{3} \) and \( \vec{a} \) is perpendicular to \( \vec{b} \). Then the greatest amongst the values of \( \lvert \vec{a} \rvert^2 \) is __________.

Let \( \vec{a}, \vec{b} \) and \( \vec{c} \) be three non zero vectors such that \( \vec{b}\!\cdot\!\vec{c} = 0 \) and \( \vec{a} \times (\vec{b} \times \vec{c}) = \dfrac{\vec{b} - \vec{c}}{2} \). If \( \vec{d} \) be a vector such that \( \vec{b}\!\cdot\!\vec{d} = \vec{a}\!\cdot\!\vec{b} \), then \( (\vec{a} \times \vec{b}) \!\cdot\! (\vec{c} \times \vec{d}) \) is equal to:

Let the position vectors of the points \( A, B, C \) and \( D \) be \( 5\hat{i} + 5\hat{j} + 2\lambda\hat{k} \), \( \hat{i} + 2\hat{j} + 3\hat{k} \), \( -2\hat{i} + \lambda\hat{j} + 4\hat{k} \) and \( -\hat{i} + 5\hat{j} + 6\hat{k} \). Let the set \( S = \{ \lambda \in \mathbb{R} : \) the points \( A, B, C \) and \( D \) are coplanar }. Then \( \sum\limits_{\lambda \in S} (\lambda + 2)^{2} \) is equal to:

Let \( \vec{a} = \hat{i} - 3\hat{j} + 7\hat{k} \), \( \vec{b} = 2\hat{i} - \hat{j} + \hat{k} \) and \( \vec{c} \) be a vector such that \( (\vec{a} + 2\vec{b}) \times \vec{c} = 3 (\vec{c} \times \vec{a}) \). If \( \vec{a} \cdot \vec{c} = 130 \), then \( \vec{b} \cdot \vec{c} \) is equal to \( \_\_\_\_\_\_\_\_\_\_\_ \).

Let \( PQR \) be a triangle such that \( \overrightarrow{PQ} = -2\hat{i} - \hat{j} + 2\hat{k} \) and \( \overrightarrow{PR} = a\hat{i} + b\hat{j} - 4\hat{k},\ a, b \in \mathbb{Z} \). Let \( S \) be the point on \( QR \), which is equidistant from the lines \( PQ \) and \( PR \). If \( |\overrightarrow{PR}| = 9 \) and \( \overrightarrow{PS} = \hat{i} - 7\hat{j} + 2\hat{k} \), then the value of \( 3a - 4b \) is ________.

Let \( \vec{a} = \sqrt{7}\,\hat{i} + \hat{j} - \hat{k} \) and \( \vec{b} = \hat{j} + 2\hat{k} \). If \( \vec{r} \) is a vector such that \( \vec{r} \times \vec{a} + \vec{a} \times \vec{b} = \vec{0} \) and \( \vec{r} \cdot \vec{a} = 0 \), then \( |3\vec{r}|^{2} \) is equal to:

A plane P contains the line of intersection of the plane \( \vec{r}\cdot(\hat{i}+\hat{j}+\hat{k}) = 6 \) and \( \vec{r}\cdot(2\hat{i}+3\hat{j}+4\hat{k}) = -5 \). If P passes through the point \( (0, 2, -2) \), then the square of distance of the point \( (12, 12, 18) \) from the plane P is:

The set of all \( \alpha \), for which the vectors \( \vec{a} = \alpha t \hat{i} + 6\hat{j} - 3\hat{k} \) and \( \vec{b} = t\hat{i} - 2\hat{j} - 2\alpha t \hat{k} \) are inclined an obtuse angle for all \( t \in \mathbb{R} \), is:

Let $\overrightarrow{AB} = 2\hat{i} + 4\hat{j} - 5\hat{k}$ and $\overrightarrow{AD} = \hat{i} + 2\hat{j} + \lambda\hat{k}, \lambda \in \mathbb{R}$. Let the projection of the vector $\vec{v} = \hat{i} + \hat{j} + \hat{k}$ on the diagonal $\overrightarrow{AC}$ of the parallelogram $ABCD$ be of length one unit. If $\alpha, \beta$, where $\alpha > \beta$, be the roots of the equation $\lambda^2x^2 - 6\lambda x + 5 = 0$, then $2\alpha - \beta$ is equal to:

Let \( \vec{a}=\hat{i}-\hat{j}+2\hat{k} \) and let \( \vec{b} \) be a vector such that \( \vec{a}\times\vec{b}=2\hat{i}-\hat{k} \) and \( \vec{a}\cdot\vec{b}=3 \). Then the projection of \( \vec{b} \) on the vector \( \vec{a}-\vec{b} \) is:

Let \( \vec{a} = 2\hat{i} + \hat{j} - 2\hat{k} \), \( \vec{b} = \hat{i} + \hat{j} \) and \( \vec{c} = \vec{a} \times \vec{b} \). Let \( \vec{d} \) be a vector such that \( \lvert \vec{d} - \vec{a} \rvert = \sqrt{11} \), \( \lvert \vec{c} \times \vec{d} \rvert = 3 \) and the angle between \( \vec{c} \) and \( \vec{d} \) is \( \frac{\pi}{4} \). Then \( \vec{a} \cdot \vec{d} \) is equal to:

Consider a \( \triangle ABC \) where \( A(1, 3, 2) \), \( B(-2, 8, 0) \) and \( C(3, 6, 7) \). If the angle bisector of \( \angle BAC \) meets the line \( BC \) at \( D \), then the length of the project of the vector \( \overrightarrow{AD} \) on the vector \( \overrightarrow{AC} \) is:

Let the three sides of a triangle \( ABC \) be given by the vectors \( 2\hat{i}-\hat{j}+ \hat{k} \), \( \hat{i}-3\hat{j}-5\hat{k} \) and \( 3\hat{i}-4\hat{j}-4\hat{k} \). Let \( G \) be the centroid of the triangle \( ABC \). Then \( 6 \left( |\overline{AG}|^2 + |\overline{BG}|^2 + |\overline{CG}|^2 \right) \) is equal to ________________.

Let \( \vec{a}=\hat{i}+\hat{j}+\hat{k} \) and \( \vec{b}=\hat{j}-\hat{k} \). If \( \vec{c} \) is a vector such that \( \vec{a}\times \vec{c}=-\vec{b} \) and \( \vec{a}\cdot \vec{c}=3 \), then \( \vec{a}\cdot (\vec{b}\times \vec{c}) \) is equal to:

Let \( \vec{c} \) be a vector perpendicular to the vectors \( \vec{a}=\hat{i}+\hat{j}-\hat{k} \) and \( \vec{b}=\hat{i}+2\hat{j}+\hat{k} \). If \( \vec{c}\cdot(\hat{i}+\hat{j}+3\hat{k})=8 \) then the value of \( \vec{c}\cdot(\vec{a}\times\vec{b}) \) is equal to ________ .

Let \( S \) be the set of all \( a \in \mathbb{R} \) for which the angle between the vectors \( \vec{u} = a(\log_{e} b)\hat{\imath} - 6\hat{\jmath} + 3\hat{k} \) and \( \vec{v} = (\log_{e} b)\hat{\imath} + 2\hat{\jmath} + 2a(\log_{e} b)\hat{k}, \ (b>1) \) is acute. Then \( S \) is equal to:

Let \( \vec{a} = 5\hat{i} - \hat{j} - 3\hat{k} \) and \( \vec{b} = \hat{i} + 3\hat{j} + 5\hat{k} \) be two vectors. Then which one of the following statements is TRUE?

Let \( \vec{a} = 9\hat{i} - 13\hat{j} + 25\hat{k} \), \( \vec{b} = 3\hat{i} + 7\hat{j} - 13\hat{k} \) and \( \vec{c} = 17\hat{i} - 2\hat{j} + \hat{k} \) be three given vectors. If \( \vec{r} \) is a vector such that \( \vec{r} \times \vec{a} = (\vec{b} + \vec{c}) \times \vec{a} \) and \( \vec{r} \cdot (\vec{b} - \vec{c}) = 0 \), then \( \dfrac{\left|593 \vec{r} + 67 \vec{a}\right|^2}{(593)^2} \) is equal to ______.

Let \( A(2,3,5) \) and \( C(-3,4,-2) \) be opposite vertices of a parallelogram \( ABCD \). If the diagonal \( \overrightarrow{BD} = \hat{i} + 2\hat{j} + 3\hat{k} \), then the area of the parallelogram is equal to:

Let \( \vec a=2\hat i-\hat j+3\hat k \), \( \vec b=3\hat i-5\hat j+\hat k \) and \( \vec c \) be a vector such that \( \vec a\times \vec c=\vec c\times \vec b \) and \( (\vec a+\vec c)\cdot(\vec b+\vec c)=168 \). Then the maximum value of \( |\vec c|^2 \) is:

Consider two vectors \( \vec{u} = 3\hat{i} - \hat{j} \) and \( \vec{v} = 2\hat{i} + \hat{j} - \lambda \hat{k}, \ \lambda > 0 \). The angle between them is given by \( \cos^{-1}\!\left(\frac{\sqrt{5}}{2\sqrt{7}}\right) \). Let \( \vec{v} = \vec{v}_1 + \vec{v}_2 \), where \( \vec{v}_1 \) is parallel to \( \vec{u} \) and \( \vec{v}_2 \) is perpendicular to \( \vec{u} \). Then the value of \( \lvert \vec{v}_1 \rvert^2 + \lvert \vec{v}_2 \rvert^2 \) is equal to:

Let $\vec{a} = \hat{i} + \hat{j} + 2\hat{k}$, $\vec{b} = 2\hat{i} - 3\hat{j} + \hat{k}$ and $\vec{c} = \hat{i} - \hat{j} + \hat{k}$ be three given vectors. Let $\vec{v}$ be a vector in the plane of $\vec{a}$ and $\vec{b}$ whose projection on $\vec{c}$ is $\frac{2}{\sqrt{3}}$. If $\vec{v} \cdot \hat{j} = 7$, $\vec{v} \cdot (\hat{i} + \hat{k})$ is equal to:

Let \( \vec{a} = \hat{i} + \hat{j} + 2\hat{k} \) and \( \vec{b} = -\hat{i} + 2\hat{j} + 3\hat{k} \). Then the vector product \( (\vec{a} + \vec{b}) \times \big((\vec{a} \times ((\vec{a} - \vec{b}) \times \vec{b})\times\vec b\big) \) is equal to:

Let $\vec{b} = \hat{i} + \hat{j} + \lambda \hat{k}, \lambda \in \mathbb{R}$. If $\vec{a}$ is a vector such that $\vec{a} \times \vec{b} = 13\hat{i} - \hat{j} - 4\hat{k}$ and $\vec{a} \cdot \vec{b} + 21 = 0$, then $(\vec{b} - \vec{a}) \cdot (\hat{k} - \hat{j}) + (\vec{b} + \vec{a}) \cdot (\hat{i} - \hat{k})$ is equal to ____________.

Let \( \overrightarrow{OA} = \vec{a} \), \( \overrightarrow{OB} = 12\vec{a} + 4\vec{b} \) and \( \overrightarrow{OC} = \vec{b} \), where \( O \) is the origin. If \( S \) is the parallelogram with adjacent sides \( OA \) and \( OC \), then \( \dfrac{\text{area of the quadrilateral } OABC}{\text{area of } S} \) is equal to _______.

In a triangle \( ABC \), if \( |\overrightarrow{BC}| = 3 \), \( |\overrightarrow{CA}| = 5 \) and \( |\overrightarrow{BA}| = 7 \), then the projection of the vector \( \overrightarrow{BA} \) on \( \overrightarrow{BC} \) is equal to:

Let \( ABC \) be a triangle of area \( 15\sqrt{2} \) and the vectors \( \overrightarrow{AB} = \hat{i} + 2\hat{j} - 7\hat{k} \), \( \overrightarrow{BC} = a\hat{i} + b\hat{j} + c\hat{k} \)  and \( \overrightarrow{AC} = 6\hat{i} + d\hat{j} - 2\hat{k},\ d > 0 \). Then the square of the length of the largest side of the triangle \( ABC \) is \( \_\_\_\_\_\_\_\_\_\_ \).

If \( (\vec{a} + 3\vec{b}) \) is perpendicular to \( (7\vec{a} - 5\vec{b}) \) and \( (\vec{a} - 4\vec{b}) \) is perpendicular to \( (7\vec{a} - 2\vec{b}) \), then the angle between \( \vec{a} \) and \( \vec{b} \) (in degrees) is ________.

A hall has a square floor of dimension \( 10 \, \text{m} \times 10 \, \text{m} \) (see the figure) and vertical walls. If the angle \(  GPH \) between the diagonals \( AG \) and \( BH \) is \( \cos^{-1}\!\left(\frac{1}{5}\right) \), then the height of the hall (in meters) is:

Let \( \vec{a} \) and \( \vec{b} \) be two non-zero vectors perpendicular to each other and \( |\vec{a}| = |\vec{b}| \). If \( |\vec{a} \times \vec{b}| = |\vec{a}| \), then the angle between the vectors \( (\vec{a} + \vec{b} +( \vec{a} \times \vec{b})) \) and \( \vec{a} \) is equal to:

Let \( \hat{a} \) and \( \hat{b} \) be two unit vectors such that the angle between them is \( \dfrac{\pi}{4} \). If \( \theta \) is the angel between the vectors \( (\hat{a}+\hat{b}) \) and \( (\hat{a}+2\hat{b}+2(\hat{a}\times\hat{b})) \), then the value of \( 164\cos^{2}\theta \) is equal to:

The least positive integral value of $\alpha$, for which the angle between the vectors $\alpha \hat{i} - 2\hat{j} + 2\hat{k}$ and $\alpha \hat{i} + 2\alpha \hat{j} - 2\hat{k}$ is acute, is ___________.

Let \( \vec{a} = \hat{i} + \hat{j} - \hat{k} \) and \( \vec{c} = 2\hat{i} - 3\hat{j} + 2\hat{k} \). Then the number of vectors \( \vec{b} \) such that \( \vec{b} \times \vec{c} = \vec{a} \) and \( |\vec{b}| \in \{1,2,\ldots,10\} \) is:

Let \( \vec{a} \) and \( \vec{b} \) be two vectors such that \( |\vec{a}| = 1 \), \( |\vec{b}| = 4 \), and \( \vec{a} \cdot \vec{b} = 2 \). If \( \vec{c} = (2\vec{a} \times \vec{b}) - 3\vec{b} \) and the angle between \( \vec{b} \) and \( \vec{c} \) is \( \alpha \), then \( 192 \sin^{2} \alpha \) is equal to __________.

Let a unit vector which makes an angle of $60^\circ$ with $2\hat{i} + 2\hat{j} - \hat{k}$ and an angle of $45^\circ$ with $\hat{i} - \hat{k}$ be $\vec{c}$. Then $\vec{c} + \left( -\frac{1}{2}\hat{i} + \frac{1}{3\sqrt{2}}\hat{j} - \frac{\sqrt{2}}{3}\hat{k} \right)$ is:

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors mutually perpendicular to each other and have same magnitude. If a vector $\vec{r}$ satisfies $\vec{a} \times \{(\vec{r} - \vec{b}) \times \vec{a}\} + \vec{b} \times \{(\vec{r} - \vec{c}) \times \vec{a}\} + \vec{c} \times \{(\vec{r} - \vec{a}) \times \vec{c}\} = \vec{0}$, then $\vec{r}$ is equal to:

If the component of \( \vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \) along and perpendicular to \( \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) respectively, are \(\displaystyle \frac{16}{11}(3\hat{i} + \hat{j} - \hat{k}) \) and \(\displaystyle \frac{1}{11}(-4\hat{i} - 5\hat{j} - 17\hat{k}) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to:

If \( \vec{a} \) and \( \vec{b} \) are two vectors such that \( |\vec{a}|=2 \) and \( |\vec{b}|=3 \), then the maximum value of \( 3\big|\!(3\vec{a}+2\vec{b})\!\big|+4\big|\!(3\vec{a}-2\vec{b})\!\big| \) is:

Let \( \vec{a} = \hat{i} + 4\hat{j} + 2\hat{k} \), \( \vec{b} = 3\hat{i} - 2\hat{j} + 7\hat{k} \) and \( \vec{c} = 2\hat{i} - \hat{j} + 4\hat{k} \). If a vector \( \vec{d} \) satisfies \( \vec{d} \times \vec{b} = \vec{c} \times \vec{b} \) and \( \vec{d} \cdot \vec{a} = 24 \), Then \( |\vec{d}|^{2} \) is equal to:

Let the vectors \( \vec{u}_{1} = \hat{i} + \hat{j} + a\hat{k} \), \( \vec{u}_{2} = \hat{i} + b\hat{j} + \hat{k} \) and \( \vec{u}_{3} = c\hat{i} + \hat{j} + \hat{k} \) be coplanar. If the vectors \( \vec{v}_{1} = (a+b)\hat{i} + c\hat{j} + c\hat{k} \), \( \vec{v}_{2} = a\hat{i} + (b+c)\hat{j} + a\hat{k} \) and \( \vec{v}_{3} = b\hat{i} + b\hat{j} + (c+a){k} \) are also coplanar, then \( 6(a+b+c) \) is equal to:

Let \( \vec{a}= \hat{\imath}+2\hat{\jmath}+3\hat{k} \), \( \vec{b}=2\hat{\imath}+3\hat{\jmath}-5\hat{k} \) and \( \vec{c}=3\hat{\imath}-\hat{\jmath}+\lambda \hat{k} \) be three vectors. Let \( \vec{r} \) unit vector along \( \vec{b}+\vec{c} \). If \( \vec{r}\cdot \vec{a}=3 \), then \( 3\lambda \) is equal to :

Let \( O \) be the origin and the position vector of the point \( P \) be \( -\hat{i}-2\hat{j}+3\hat{k} \). If the position vectors of the points \( A \), \( B \) and \( C \) are \( -2\hat{i}+\hat{j}-3\hat{k} \), \( 2\hat{i}+4\hat{j}-2\hat{k} \) and \( -4\hat{i}+2\hat{j}-\hat{k} \) respectively, then the projection of the vector \( \overrightarrow{OP} \) on a vector perpendicular to the vectors \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \) is:

If the points with position vectors \( \alpha \hat{i}+10\hat{j}+13\hat{k},\; 6\hat{i}+11\hat{j}+11\hat{k},\; \dfrac{9}{2}\hat{i}+\beta \hat{j}-8\hat{k} \) are collinear, then \( (19\alpha-6\beta)^2 \) is equal to:

Let \( A \) and \( B \) be two distinct points on the line \( L : \frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2} \). Both \( A \) and \( B \) are at a distance \( 2\sqrt{17} \) from the foot of perpendicular drawn from the point \( (1,2,3) \) on the line \( L \). If \( O \) is the origin, then \( \overrightarrow{OA}\cdot\overrightarrow{OB} \) is equal to:

Let \( \vec{a} \) be a vector which is perpendicular to the vector \( 3\hat{i} + \tfrac{1}{2}\hat{j} + 2\hat{k} \). If \( \vec{a} \times (2\hat{i} + \hat{k}) = 2\hat{i} - 13\hat{j} - 4\hat{k} \), then the projection of the vector \( \vec{a} \) on the vector \( 2\hat{i} + 2\hat{j} + \hat{k} \) is:

Let \( \vec{a}=2\hat{i}-\hat{j}-\hat{k},\ \vec{b}=\hat{i}+3\hat{j}-\hat{k} \) and \( \vec{c}=2\hat{i}+\hat{j}+3\hat{k} \). Let \( \vec{v} \) be the vector in the plane of the vectors \( \vec{a} \) and \( \vec{b} \), such that the length of its projection on the vector \( \vec{c} \) is \( \frac{1}{\sqrt{14}} \). Then \( |\vec{v}| \) is equal to:

Let \( \vec{a} = \hat{\imath} + \alpha \hat{\jmath} + \beta \hat{k}, \ \alpha,\beta \in \mathbb{R} \). Let a vector \( \vec{b} \) be such that the angle between \( \vec{a} \) and \( \vec{b} \) is \( \dfrac{\pi}{4} \) and \(|\vec{b}|^{2} = 6 \). If \( \vec{a} \cdot \vec{b} = 3\sqrt{2} \), then the value of \( (\alpha^{2} + \beta^{2}) \, |\vec{a} \times \vec{b}|^{2} \) is equal to:

Let $O$ be the origin and the position vectors of $A$ and $B$ be $2\hat{i} + 2\hat{j} + \hat{k}$ and $2\hat{i} + 4\hat{j} + 4\hat{k}$ respectively. If the internal bisector of $\angle AOB$ meets the line $AB$ at $C$, then the length of $OC$ is:

Let \( \overrightarrow{OA}=2\vec{a},\ \overrightarrow{OB}=6\vec{a}+5\vec{b} \) and \( \overrightarrow{OC}=3\vec{b} \), where \( O \) is the origin. If the area of the parallelogram with adjacent sides \( \overrightarrow{OA} \) and \( \overrightarrow{OC} \) is \( 15 \) sq. units, then the area (in sq. units) of the quadrilateral \( OABC \) is equal to:

Let the position vectors of the vertices $A, B$ and $C$ of a tetrahedron $ABCD$ be $\hat{i} + 2\hat{j} + \hat{k}$, $\hat{i} + 3\hat{j} - 2\hat{k}$ and $2\hat{i} + \hat{j} - \hat{k}$ respectively. The altitude from the vertex $D$ to the opposite face $ABC$ meets the median line segment through $A$ of the triangle $ABC$ at the point $E$. If the length of $AD$ is $\frac{\sqrt{110}}{3}$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6\sqrt{2}}$, then the position vector of $E$ is:

The shortest distance between the lines \( \vec{r}=\left(\frac{1}{3}\hat{i}+2\hat{j}+\frac{8}{3}\hat{k}\right)+\lambda(2\hat{i}-5\hat{j}+6\hat{k}) \) and \( \vec{r}=\left(-\frac{2}{3}\hat{i}-\frac{1}{3}\hat{k}\right)+\mu(\hat{j}-\hat{k}), \lambda,\mu \in \mathbb{R}, \) is:

Let the position vectors of vertices \( A, B \) and \( C \) of a triangle be \( 2\hat{i} + 2\hat{j} + \hat{k}, \, \hat{i} + 2\hat{j} + 2\hat{k} \),and \( 2\hat{i} + \hat{j} + 2\hat{k}\) respectively. Let \( l_1, l_2 \) and \( l_3 \) be the lengths of perpendiculars drawn from the ortho centre of the triangle on the sides \( AB, BC \) and \( CA \) respectively, then \( l_1^2+l_2^2+l_3^2 \) equals:

If the vectors \( \vec{a} = \lambda \hat{i} + \mu \hat{j} + 4\hat{k} \), \( \vec{b} = -2\hat{i} + 4\hat{j} - 2\hat{k} \) and \( \vec{c} = 2\hat{i} + 3\hat{j} + \hat{k} \) are coplanar and the projection of \( \vec{a} \) on the vector \( \vec{b} \) is \( \sqrt{54} \) units, then the sum of all possible values of \( \lambda + \mu \) is equal to:

If \( \vec{a}=\hat{i}+\hat{j}+\hat{k} \), \( \vec{b}=\hat{j}-\hat{k} \) and \( \vec{c} \) be three vectors such that \( \vec{a}\times\vec{c}=\vec{b} \) and \( \vec{a}\cdot\vec{c}=3 \), then \( \vec{c}\cdot(\vec{a}-2\vec{b}) \) is equal to ______.

The square of the distance of the point \( P(5, 6, 7) \) from the line \( \dfrac{x-2}{2} = \dfrac{y-5}{3} = \dfrac{z-2}{4} \) is equal to:

Let \( \vec{a} = 3\hat{i} - \hat{j} + 2\hat{k} \), \( \vec{b} = \vec{a} \times (\hat{i} - 2\hat{k}) \) and \( \vec{c} = \vec{b} \times \hat{k} \). Then the projection of \( \vec{c} - 2\hat{j} \) on \( \vec{a} \) is:

Let \( \vec{a}=\hat{i}+\hat{j}-\hat{k},\ \vec{b} \) and \( \vec{c}=\hat{j}-\hat{k} \) be three vectors such that \( \vec{a}\times\vec{b}=\vec{c} \) and \( \vec{a}\cdot\vec{b}=1 \). If the length of projection vector of the vector \( \vec{b} \) on the vector \( \vec{a}\times\vec{c} \) is \( l \) then the value of \( 3l^{2} \) is equal to ________.

Let \( \vec{a}=\hat{i}-\alpha\hat{j}+\beta\hat{k},\ \vec{b}=3\hat{i}+\beta\hat{j}-\alpha\hat{k} \) and \( \vec{c}=-\alpha\hat{i}-2\hat{j}+\hat{k} \), where \( \alpha \) and \( \beta \) are integers. If \( \vec{a}\cdot\vec{b}=-1 \) and \( \vec{b}\cdot\vec{c}=10 \), then \( (\vec{a}\times\vec{b})\cdot\vec{c} \) is equal to __________.

Two vectors \( \vec{X} \) and \( \vec{Y} \) have equal magnitude. The magnitude of \( (\vec{X}-\vec{Y}) \) is \( n \) times the magnitude of \( (\vec{X}+\vec{Y}) \). The angle between \( \vec{X} \) and \( \vec{Y} \) is:

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|2\vec{a} + 3\vec{b}| = |3\vec{a} + \vec{b}|$ and the angle between $\vec{a}$ and $\vec{b}$ is $60^\circ$. If $\frac{1}{8}\vec{a}$ is a unit vector, then $|\vec{b}|$ is equal to:

Let a vector \( \vec{a} = \sqrt{2}\,\hat{i} -\hat{j} + \lambda \hat{k};\; \lambda > 0 \), make an obtuse angle with the vector \( \vec{b} = -\lambda^{2}\hat{i} + 4\sqrt{2}\,\hat{j} + 4\sqrt{2}\,\hat{k} \) and an angle \( \theta,\; \frac{\pi}{6} < \theta < \frac{\pi}{2} \) with the positive z-axis. If the set of all possible values of \( \lambda \) is \( (\alpha,\beta) - \{\gamma\} \) then \( \alpha + \beta + \gamma \) is equal to ______ .

Let \( \vec{a}=4\hat{i}-\hat{j}+\hat{k} \) and \( \vec{b}=11\hat{i}-\hat{j}-\hat{k} \) and \( \vec{c} \) be a vector such that \( (\vec{a}+\vec{b})\times\vec{c}=\vec{c}\times(2\vec{a}+3\vec{b}) \). If \( (2\vec{a}+3\vec{b}).\\vec{c}=1670 \), then \( |\vec{c}|^2 \) is equal to:

For any vector \( \vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k} \), with \( 10|a_i|<1, \ i=1,2,3 \), consider the following statements:
(A) : \( \max\{|a_1|,|a_2|,|a_3|\}\le |\vec{a}| \)
(B) : \( |\vec{a}|\le 3\max\{|a_1|,|a_2|,|a_3|\} \)

Let \( \vec{a}=\hat{i}+2\hat{j}+\hat{k} \) and \( \vec{b}=2\hat{i}+\hat{j}-\hat{k} \). Let \( \hat{c} \) be a unit vector in the plane of the vectors \( \vec{a} \) and \( \vec{b} \) and be perpendicular to \( \vec{a} \). Then such a vector \( \hat{c} \) is:

Let \( \theta \) be the angle between the vectors \( \vec{a} \) and \( \vec{b} \), where \( |\vec{a}| \ = 4 \), \(  |\vec{b}| = 3 \) and \( \theta \in \left( \dfrac{\pi}{4}, \dfrac{\pi}{3} \right) \).
Then \( | (\vec{a}-\vec{b}) \times (\vec{a}+\vec{b}) |^{2} + 4(\vec{a}\cdot\vec{b})^{2} \) is equal to ______.

If the points \( P \) and \( Q \) are respectively the circumcenter and the orthocenter of a \( \triangle ABC \), then \( \overrightarrow{PA}+\overrightarrow{PB}+\overrightarrow{PC} \) is:

If the shortest distance between the lines $\vec{r} = (-\hat{i} + 3\hat{k}) + \lambda(\hat{i} - a\hat{j})$ and $\vec{r} = (-\hat{j} + 2\hat{k}) + \mu(\hat{i} - \hat{j} + \hat{k})$ is $\sqrt{\frac{2}{3}}$, then the integral value of $a$ is equal to __________.

If $\vec{a} = 2\hat{i} + \hat{j} + 3\hat{k}$, $\vec{b} = 3\hat{i} + 3\hat{j} + \hat{k}$ and $\vec{c} = c_1\hat{i} + c_2\hat{j} + c_3\hat{k}$ are coplanar vectors and $\vec{a} \cdot \vec{c} = 5$, $\vec{b} \perp \vec{c}$, then $122(c_1 + c_2 + c_3)$ is equal to ______________.

Let \( \hat{a} \) be a unit vector perpendicular to the vectors \( \vec{b} = \hat{i} - 2\hat{j} + 3\hat{k} \) and \( \vec{c} = 2\hat{i} + 3\hat{j} - \hat{k} \), and makes an angle of \(\displaystyle 
\cos^{-1}\!\left(-\frac{1}{3}\right) \) with the vector \( \hat{i} + \hat{j} + \hat{k} \). If \( \hat{a} \) makes an angle of \(\displaystyle
 \frac{\pi}{3} \) with the vector \( \hat{i} + \alpha \hat{j} + \hat{k} \), then the value of \( \alpha \) is:

Let \( \vec{a},\vec{b},\vec{c} \) be three non-coplanar vectors such that \( \vec{a}\times \vec{b}=4\vec{c},\, \vec{b}\times \vec{c}=9\vec{a} \) and \( \vec{c}\times \vec{a}=\alpha \vec{b} \) and \( \alpha>0 \). If \( |\vec{a}|+|\vec{b}|+|\vec{c}|=\dfrac{1}{36} \), then \( \alpha \) is equal to ________.

Let \( \vec{a}=\alpha \hat{i}+3\hat{j}-\hat{k}, \ \vec{b}=3\hat{i}-\beta \hat{j}+4\hat{k} \) and \( \vec{c}=\hat{i}+2\hat{j}-2\hat{k} \) where \( \alpha,\beta \in R \), be three vectors. If the projection of \( \vec{a} \) on \( \vec{c} \) is \( \dfrac{10}{3} \) and \( \vec{b} \times \vec{c}=-6\hat{i}+10\hat{j}+7\hat{k} \), then the value of \( \alpha+\beta \) is equal to:

\( A(2,6,2), \; B(-4,0,\lambda), \; C(2,3,-1) \) and \( D(4,5,0) \), \( |\lambda| \le 5 \) are the vertices of a quadrilateral \( ABCD \). If its area is \( 18 \) square units, then \( 5 - 6\lambda \) is equal to ________.