JEE MAIN Vector Previous Year Questions (PYQs) – Page 10 of 17

The distance of the point $P(4,6,-2)$ from the line passing through the point $(-3,2,3)$ and parallel to a line with direction ratios $3,3,-1$ is equal to:

The area (in sq. units) of the parallelogram whose diagonals are along the vectors $8\hat{i}-6\hat{j}$ and $3\hat{i}+4\hat{j}-12\hat{k}$, is :

Consider the lines $L_{1}$ and $L_{2}$ given by $L_{1}:\ \dfrac{x-1}{2}=\dfrac{y-3}{1}=\dfrac{z-2}{2}$ $L_{2}:\ \dfrac{x-2}{1}=\dfrac{y-2}{2}=\dfrac{z-3}{3}$ A line $L_{3}$ having direction ratios $1,-1,-2$ intersects $L_{1}$ and $L_{2}$ at the points $P$ and $Q$ respectively. Then the length of line segment $PQ$ is:

Let $\vec a=3\hat{i}+2\hat{j}+x\hat{k}$ and $\vec b=\hat{i}-\hat{j}+\hat{k}$, for some real $x$. Then $\left|\vec a\times\vec b\right|=r$ is possible if:

Let $L_1:\ \vec r=(\hat i-\hat j+2\hat k)+\lambda(\hat i-\hat j+2\hat k),\ \lambda\in\mathbb R,$ $L_2:\ \vec r=(\hat j-\hat k)+\mu(3\hat i+\hat j+p\hat k),\ \mu\in\mathbb R,$ and $L_3:\ \vec r=\delta(\ell\hat i+m\hat j+n\hat k),\ \delta\in\mathbb R,$ be three lines such that $L_1$ is perpendicular to $L_2$ and $L_3$ is perpendicular to both $L_1$ and $L_2$. Then, the point which lies on $L_3$ is:

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

Let $\vec a=\hat i+\alpha\hat j+\beta\hat k,\ \alpha,\beta\in\mathbb R$. Let $\vec b$ 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:

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

The lines
$\overrightarrow r = \left( {\widehat i - \widehat j} \right) + l\left( {2\widehat i + \widehat k} \right)$ and
$\overrightarrow r = \left( {2\widehat i - \widehat j} \right) + m\left( {\widehat i + \widehat j + \widehat k} \right)$

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