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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:

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JEE Main 2022 29th July Evening shift

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: