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The position vectors of the vertices \( A, B \) and \( C \) of a triangle are \( 2\hat{i}-3\hat{j}+3\hat{k} \), \( 2\hat{i}+2\hat{j}+3\hat{k} \) and \( -\hat{i}+\hat{j}+3\hat{k} \) respectively. Let \( l \) denotes the length of the angle bisector \( AD \) of \( \angle BAC \) where \( D \) is on the line segment \( BC \), then \( 2l^{2} \) equals:

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 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

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

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

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 \( \vec{a} = 2\hat{i} + \alpha \hat{j} + \hat{k}, \ \vec{b} = -\hat{i} + \hat{k}, \ \vec{c} = \beta \hat{j} - \hat{k} \), 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:

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:

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:

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 \( \vec{a} = \hat{i} + \hat{j} + \hat{k} \), \( \vec{b} = -\hat{i} - 8\hat{j} + 2\hat{k} \) 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 \( \vec{a} = \hat{i} + \hat{j} + \hat{k}, \\ \vec{b} = 2\hat{i} + 4\hat{j} - 5\hat{k} \) 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 \( \vec{a}=\hat{i}+2\hat{j}+\hat{k} \), \( \vec{b}=3(\hat{i}-\hat{j}+\hat{k}) \). 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:

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 $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ 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:

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 \( \vec{a} = \hat{i} - 3\hat{j} + 7\hat{k} \), \( \vec{b} = 2\hat{i} - \hat{j} + \hat{k} \) 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 \( \_\_\_\_\_\_\_\_\_\_\_ \).

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:

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 \( \vec{a} = 9\hat{i} - 13\hat{j} + 25\hat{k} \), \( \vec{b} = 3\hat{i} + 7\hat{j} - 13\hat{k} \) 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 \( \overrightarrow{OA} = \vec{a} \), \( \overrightarrow{OB} = 12\vec{a} + 4\vec{b} \) 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 _______.

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 \( \_\_\_\_\_\_\_\_\_\_ \).

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 \( \vec{a} \) 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 \( \vec{a}= \hat{\imath}+2\hat{\jmath}+3\hat{k} \), \( \vec{b}=2\hat{\imath}+3\hat{\jmath}-5\hat{k} \) 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 \( \vec{a} = \hat{\imath} + \alpha \hat{\jmath} + \beta \hat{k}, \ \alpha,\beta \in \mathbb{R} \). 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 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:

Let \( \vec{a}, \vec{b} \) and \( \vec{c} \) be three non-zero vectors such that \( \vec{b} \) and \( \vec{c} \) are non-collinear. If \( \vec{a} + 5\vec{b} \) is collinear with \( \vec{c} \), \( \vec{b} + 6\vec{c} \) is collinear with \( \vec{a} \) and \( \vec{a} + \alpha \vec{b} + \beta \vec{c} = \vec{0} \), then \( \alpha + \beta \) 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: