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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:

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:

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 \( \vec{a}=3\hat{i}+\hat{j} \) 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:

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 \( \vec{a} \) 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 \( \hat{a} \) 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:

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

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:

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:

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 \(  ABC \) 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

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 $\vec{a} = \alpha\hat{i} + 2\hat{j} - \hat{k}$ 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 \( \vec{a}=a_{1}\hat{i}+a_{2}\hat{j}+a_{3}\hat{k},\ a_{i}>0,\ i=1,2,3 \) 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:

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 \( \vec{a}=\hat{i}-\hat{j}+2\hat{k} \) 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 \( S \) 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 $\vec{a} = \hat{i} + \hat{j} + 2\hat{k}$, $\vec{b} = 2\hat{i} - 3\hat{j} + \hat{k}$ 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 $\vec{b} = \hat{i} + \hat{j} + \lambda \hat{k}, \lambda \in \mathbb{R}$. 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 \( \hat{a} \) 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:

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