JEE Main 2022 — Vectors & 3D Geometry

Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that the angle between them is $\frac{\pi }{4}$. If $\theta$ is the angle between the vectors $(\hat{a}+\hat{b})$ and $(\hat{a}+2\hat{b}+2(\hat{a}\times \hat{b}))$ then the value of $164{\mathrm{cos}}^{2}\theta$ is equal to

Let $Q$ and $R$ be two points on the line $\frac{x+1}{2}=\frac{y+2}{3}=\frac{z-1}{2}$ at a distance $\sqrt{26}$ from the point $P(4,2,7)$. Then the square of the area of the triangle $PQR$ is ________.

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 the 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 $\vec{v},\hat{j}=7$, then $\vec{v}\cdot (\hat{i}+\hat{k})$ is equal to

If the two lines ${l}_{1}:\frac{x-2}{3}=\frac{y+1}{-2},z=2$ and ${l}_{2}:\frac{x-1}{1}=\frac{2y+3}{\alpha }=\frac{z+5}{2}$ are perpendicular, then an angle between the lines ${l}_{2}$ and ${l}_{3}:\frac{1-x}{3}=\frac{2y-1}{-4}=\frac{z}{4}$ is

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 ______.

The shortest distance between the lines $\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-1}{-1}$ and $\frac{x+3}{2}=\frac{y-6}{1}=\frac{z-5}{3}$ is

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 $\hat{a},\hat{b}$ 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 ${|6\vec{c}|}^{2}$ is equal to:

Let ${l}_{1}$ be the line in $xy$-plane with $x$ and $y$ intercepts $\frac{1}{8}$ and $\frac{1}{4\sqrt{2}}$ respectively, and ${l}_{2}$ be the line in $zx$-plane with $x$ and $z$ intercepts $-\frac{1}{8}$ and $-\frac{1}{6\sqrt{3}}$ respectively. If $d$ is the shortest distance between the line ${l}_{1}$ and ${l}_{2}$, then ${d}^{-2}$ is equal to _____.

Let $ABC$ be a triangle such that $\vec{BC}=\vec{a},\vec{CA}=\vec{b}$, $\vec{AB}=\vec{c},|\vec{a}|=6\sqrt{2},|\vec{b}|=2\sqrt{3}$ and $\vec{b}\cdot \vec{c}=12$ Consider the statements : $(S1)$: $|(\vec{a}\times \vec{b})+(\vec{c}\times \vec{b})|-|\vec{c}|=6(2\sqrt{2}-1)$ $(S2)$: $\angle ABC={\mathrm{cos}}^{-1}(\sqrt{\frac{2}{3}})$. Then

If the shortest distance between the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{\lambda }$ and $\frac{x-2}{1}=\frac{y-4}{4}=\frac{z-5}{5}$ is $\frac{1}{\sqrt{3}}$, then the sum of all possible values of $\lambda$ is:

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 $\frac{10}{3}$ and$\vec{b}\times \vec{c}=-6\hat{i}+10\hat{j}+7\hat{k}$ , then the value of $\alpha +\beta$ 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, such that $\vec{a}\times \vec{b}=-\hat{i}+9\hat{i}+12\hat{k}$. Then the projection of $\vec{b}-2\vec{a}$ on $\vec{b}+\vec{a}$ is equal to

Let $A,B,C$ be three points whose position vectors respectively are: $\vec{a}=\hat{i}+4\hat{j}+3\hat{k}$ $\vec{b}=2\hat{i}+\alpha \hat{j}+4\hat{k},\alpha \in R$ $\vec{c}=3\hat{i}-2\hat{j}+5\hat{k}$ If $\alpha$ is the smallest positive integer for which $\vec{a},\vec{b},\vec{c}$ are non-collinear, then the length of the median, $\triangle ABC$, through $A$ is:

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 $\vec{a}=\hat{i}+\hat{j}-\hat{k}$ 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 $\vec{a}=\hat{i}-2\hat{j}+3\hat{k},\vec{b}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that $\vec{a}\times (\vec{b}+\vec{c})=\vec{0}$, then the value of $3(\vec{c}.\vec{a})$ is equal to _______.

The line of shortest distance between the lines $\frac{x-2}{0}=\frac{y-1}{1}=\frac{z}{1}$ and $\frac{x-3}{2}=\frac{y-5}{2}=\frac{z-1}{1}$ makes an angle of ${\mathrm{sin}}^{-1}(\sqrt{\frac{2}{27}})$ with the plane $P:ax-y-z=0,(a>0)$. If the image of the point $(1,1,-5)$ in the plane $P$ is $(\alpha ,\beta ,\gamma )$, then $\alpha +\beta -\gamma$ is equal to _____ .

Let $P(-2,-1,1)$ and $Q(\frac{56}{17},\frac{43}{17},\frac{111}{17})$ be the vertices of the rhombus $PRQS$. If the direction ratios of the diagonal $RS$ are $\alpha ,-1,\beta$, where both $\alpha$ and $\beta$ are integers of minimum absolute values, then ${\alpha }^{2}+{\beta }^{2}$ is equal to

Let $S$ be the set of all $a\in R$ for which the angle between the vectors $\vec{u}=a({\mathrm{log}}_{e}b)\hat{i}-6\hat{j}+3\hat{k}$ and $\vec{v}=({\mathrm{log}}_{e}b)\hat{i}+2\hat{j}+2a({\mathrm{log}}_{e}b)\hat{k},(b>1)$ is acute. Then $S$ is equal to

Let a vector $\vec{a}$ has a magnitude $9$. Let a vector $\vec{b}$ be such that for every $(x,y)R\times R-{(0,0)}$, the vector $(x\vec{a}+y\vec{b})$ is perpendicular to the vector $(6y\vec{a}-18x\vec{b})$. Then the value of $|\vec{a}\times \vec{b}|$ is equal to

If the length of the perpendicular drawn from the point $P(a,4,2),a>0$ on the line $\frac{x+1}{2}=\frac{y-3}{3}=\frac{z-1}{-1}$ is $2\sqrt{6}$ units and $Q({\alpha }_{1},{\alpha }_{2},{\alpha }_{3})$ is the image of the point $P$ in this line, then $a+\sum _{i=1}^{3}{\alpha }_{i}$ 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 two straight lines whose direction cosines are given by the relations $l+m-n=0,3{l}^{2}+{m}^{2}+cnl=0$ are parallel, then the positive value of $c$ is

Let a line having direction ratios $1,-4,2$ intersect the lines $\frac{x-7}{3}=\frac{y-1}{-1}=\frac{z+2}{1}$ and $\frac{x}{2}=\frac{y-7}{3}=\frac{z}{1}$ at the points $A$ and $B$. Then ${(AB)}^{2}$ is equal to

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

The shortest distance between lines (x-1)/2=(y+1)/3=z/1 and (x+1)/5=(y-2)/1=z/(-1) is:

Let $\vec{a}$ be a vector which is perpendicular to the vector $3\hat{i}+\frac{1}{2}\hat{j}+2\hat{k}$. 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 $\vec{a}=\alpha \hat{i}+2\hat{j}-\hat{k}$ and $\vec{b}=-2\hat{i}+\alpha \hat{j}+\hat{k}$, where $\alpha \in R$. 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 $\vec{a}=\hat{i}-\hat{j}+2\hat{k}$ 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:

The shortest distance between the lines $\frac{x+7}{-6}=\frac{y-6}{7}=z$ and $\frac{7-x}{2}=y-2=z-6$ is

The direction cosines of the line joining points (1,2,3) and (4,6,3) are:

Let $\theta$ be the angle between the vectors $\vec{a}$ and $\vec{b}$, where $|\vec{a}|=4,|\vec{b}|=3$ and $\theta \in (\frac{\pi }{4},\frac{\pi }{3})$. Then ${|(\vec{a}-\vec{b})\times (\vec{a}+\vec{b})|}^{2}+4{(\vec{a}\cdot \vec{b})}^{2}$ is equal to ______

Let $\vec{a}$ and $\vec{b}$ be the vectors along the diagonal of a parallelogram having area $2\sqrt{2}$. Let the angle between $\vec{a}$ and $\vec{b}$ be acute. $|\vec{a}|=1$ and $|\vec{a}.\vec{b}|=|\vec{a}\times \vec{b}|$. 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 $\vec{a}={a}_{1}\hat{i}+{a}_{2}\hat{j}+{a}_{3}\hat{k},{a}_{i}>0,i=1,2,3$ 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 a vector $\vec{b}$ on $3\hat{i}+4\hat{j}$ 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 $((\vec{a}\times \vec{b})\times \hat{i})\cdot \hat{k}=\frac{23}{2}$, then $|\vec{b}\times 2\hat{j}|$ is equal to

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 $\vec{c}=t\hat{i}-t\hat{j}+\hat{k},t\in R$ be such that for $\alpha ,\beta ,\gamma \in R,\alpha \vec{a}+\beta \vec{b}+\gamma \vec{c}=\vec{0}$ $\Rightarrow \alpha =\beta =\gamma =0$. Then, the set of all values of $t$ is

Let $a$ and $b$ be two unit vectors such that $|(a+b)+2(a\times b)|=2$. If $\theta \in (0,\pi )$ is the angle between $\hat{a}$ and $\hat{b}$, then among the statements: $(S1):2|\hat{a}\times \hat{b}|=|\hat{a}-\hat{b}|$ $(S2)$ : The projection of $\hat{a}$ on $(\hat{a}+\hat{b})$ is $\frac{1}{2}$