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Let \( \vec{a} \) be a non-zero vector parallel to the line of intersection of the two planes described by \( \hat{i}+\hat{j}, \hat{i}+\hat{k} \) and \( \hat{i}-\hat{j}, \hat{j}-\hat{k} \). If \( \theta \) is the angle between the vector \( \vec{a} \) and the vector \( \vec{b}=2\hat{i}-2\hat{j}+\hat{k} \) and \( \vec{a}\cdot\vec{b}=6 \), then the ordered pair \( \left(\theta, \left|\vec{a}\times\vec{b}\right|\right) \) is equal to:

Let: \( \vec{a}=\hat{i}+2\hat{j}+3\hat{k} \), \( \vec{b}=\hat{i}-\hat{j}+2\hat{k} \) and \( \vec{c}=5\hat{i}-3\hat{j}+3\hat{k} \) be vectors. If \( \vec{r} \) is a vector such that, \( \vec{r}\times\vec{b}=\vec{c}\times\vec{b} \) and \( \vec{r}\cdot\vec{a}=0 \), then \( 25|\vec{r}|^{2} \) is equal to:

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:

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 \( \vec{a},\vec{b},\vec{c} \) 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 \( \vec{u} = \hat{i} - \hat{j} - 2\hat{k} \), \( \vec{v} = 2\hat{i} + \hat{j} - \hat{k} \), \( \vec{v} \cdot \vec{w} = 2 \) 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:

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:

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:

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 \( \vec{a} = 4\hat{i} + 3\hat{j} \) 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 \( ABCD \) 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 \( \vec{a}=-\hat{i}-\hat{j}+\hat{k} \), \( \vec{a}\cdot\vec{b}=1 \) 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:

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:

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:

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 \( \vec{b} \) 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 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 \( \vec{a}, \ \vec{b} \) 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 \( \vec a=2\hat i-7\hat j+5\hat k \), \( \vec b=\hat i+\hat k \) 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:

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 ________.

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 \( \vec{a}, \vec{b} \) 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:

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:

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 \( \vec{a} = \hat{i} + 4\hat{j} + 2\hat{k} \), \( \vec{b} = 3\hat{i} - 2\hat{j} + 7\hat{k} \) 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 \( O \) 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:

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:

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) : \( |\vec{a}|\le 3\max\{|a_1|,|a_2|,|a_3|\} \)

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:

\( A(2,6,2), \; B(-4,0,\lambda), \; C(2,3,-1) \) 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 ________.

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JEE Main 2023 1st Feb Morning Shift

\( A(2,6,2), \; B(-4,0,\lambda), \; C(2,3,-1) \) 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 ________.