JEE Main 2023 — Vectors & 3D Geometry

An arc $PQ$ of a circle subtends a right angle at its centre $O$. The mid point of the arc $PQ$ is $R$. If $\vec{OP}=\vec{u},\vec{OR}=\vec{v}$ and $\vec{OQ}=\alpha \vec{u}+\beta \vec{v}$, then $\alpha ,{\beta }^{2}$, are the roots of the equation

$A(2,6,2),B(-4,0,\lambda ),C(2,3,-1)$ and $D(4,5,0),|\lambda |\leq 5$ are the vertices of a quadrilateral $ABCD$. If its area is $18$ square units, then $5-6\lambda$ is equal to _____ .

Consider the lines ${L}_{1}$ and ${L}_{2}$ given by ${L}_{1}:\frac{x-1}{2}=\frac{y-3}{1}=\frac{z-2}{2}$ ${L}_{2}:\frac{x-2}{1}=\frac{y-2}{2}=\frac{z-3}{3}$ 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

For any vector $\vec{a}={a}_{1}\hat{i}+{a}_{2}\hat{j}+{a}_{3}\hat{k}$, with $10|{a}_{i}|<1,i=1,2,3$, consider the following statements: $(A)$ : $\mathrm{max}{|{a}_{1}|,|{a}_{2}|,|{a}_{3}|}\leq |\vec{a}|$ $(B)$ : $|\vec{a}|\leq 3\mathrm{max}{|{a}_{1}|,|{a}_{2}|,|{a}_{3}|}$

If $\vec{a}=\hat{i}+2\hat{k},\vec{b}=\hat{i}+\hat{j}+\hat{k},\vec{c}=7\hat{i}-3\hat{j}+4\hat{k},\vec{r}\times \vec{b}+\vec{b}\times \vec{c}=\vec{0}$ and $\vec{r}\cdot \vec{a}=0$ then $\vec{r}.\vec{c}$ is equal to:

If the line $x=y=z$ intersects the line $x\mathrm{sin}A+y\mathrm{sin}B+z\mathrm{sin}C-18=0=x\mathrm{sin}2A+y\mathrm{sin}2B+z\mathrm{sin}2C-9$, where $A,B,C$ are the angles of a triangle $ABC$, then $80(\mathrm{sin}\frac{A}{2}\mathrm{sin}\frac{B}{2}\mathrm{sin}\frac{C}{2})$ is equal to _________.

If the lines $\frac{x-1}{2}=\frac{2-y}{-3}=\frac{z-3}{\alpha }$ and $\frac{x-4}{5}=\frac{y-1}{2}=\frac{z}{\beta }$ intersect, then the magnitude of the minimum value of $8\alpha \beta$ is _____.

If the lines $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z+3}{1}$ and $\frac{x-a}{2}=\frac{y+2}{3}=\frac{z-3}{1}$ intersects at the point $P$, then the distance of the point $P$ from the plane $z=a$ is :

If the points with position vectors $\alpha \hat{i}+10\hat{j}+13\hat{k},6\hat{i}+11\hat{j}+11\hat{k},\frac{9}{2}\hat{i}+\beta \hat{j}-8\hat{k}$ are collinear, then ${(19\alpha -6\beta )}^{2}$ is equal to

If the shortest distance between the line joining the points $(1,2,3)$ and $(2,3,4)$, and the line $\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-2}{0}$ is $\alpha$, then $28{\alpha }^{2}$ is equal to _____ .

If the shortest distance between the lines $\frac{x+\sqrt{6}}{2}=\frac{y-\sqrt{6}}{3}=\frac{z-\sqrt{6}}{4}$ and $\frac{x-\lambda }{3}=\frac{y-2\sqrt{6}}{4}=\frac{z+2\sqrt{6}}{5}$ is $6$, then sum of squares of all possible values(s) of $\lambda$ is

If the vectors $\vec{a}=\lambda \hat{i}+\mu \hat{j}+4\hat{k},\vec{b}=-2\hat{i}+4\hat{j}-2\hat{k}$ and $\vec{c}=2\hat{i}+3\hat{j}+\hat{k}$ are coplanar and the projection of $\vec{a}$ on the vector $\vec{b}$ is $\sqrt{54}$ units, then the sum of all possible values of $\lambda +\mu$ is equal to

The equation of the plane passing through (1,2,3) and perpendicular to the vector 2i+3j-k is:

The scalar triple product [i j k] is:

Let a line $L$ pass through the origin and be perpendicular to the lines ${L}_{1}:\vec{r}=(\hat{i}-11\hat{j}-7\hat{k})+\lambda (\hat{i}+2\hat{j}+3\hat{k}),\lambda \in \mathbb{R}$ and ${L}_{2}:\vec{r}=(-\hat{i}+\hat{k})+\mu (2\hat{i}+2\hat{j}+\hat{k}),\mu \in \mathbb{R}$. If $P$ is the point of intersection of $L$ and ${L}_{1}$, and ,Q\alpha ,\beta ,\gamma is the foot of perpendicular from $P$ on ${L}_{2}$, then $9(\alpha +\beta +\gamma )$ is equal to ________.

Let a line $L$ pass through the point $P(2,3,1)$ and be parallel to the line $x+3y-2z-2=0=x-y+2z$. If the distance of $L$ from the point $(5,3,8)$ is $\alpha$, then $3{\alpha }^{2}$ is equal to ________

Let $\vec{a}=2\hat{i}+\hat{j}+\hat{k}$, and $\vec{b}$ and $\vec{c}$ be two nonzero vectors such that $|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}|$ and $\vec{b}\cdot \vec{c}=0$. Consider the following two statement: $(A)|\vec{a}+\lambda \vec{c}|\geq |\vec{a}|$ for all $\lambda \in \mathbb{R}$. $(B)$ $\vec{a}$ and $\vec{c}$ are always parallel

Let $\vec{a}=4\hat{i}+3\hat{j}$ and $\vec{b}=3\hat{i}-4\hat{j}+5\hat{k}$ and $\vec{c}$ is a vector such that $\vec{c}\cdot (\vec{a}\times \vec{b})+25=0,\vec{c}\cdot (\hat{i}+\hat{j}+\hat{k})=4$ and projection of $\vec{c}$ on $\vec{a}$ is $1$ , then the projection of $\vec{c}$ on $\vec{b}$ equals:

Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k},\vec{b}=\hat{i}-\hat{j}+2\hat{k}$ and $\vec{c}=5\hat{i}-3\hat{j}+3\hat{k}$, be there(three) vector. If $\vec{r}$ is a vector such that, $\vec{r}\times \vec{b}=\vec{c}\times \vec{b}$ and $\vec{r}\cdot \vec{a}=0$, then $25{|\vec{r}|}^{2}$ is equal to

Let $\vec{a}=2\hat{i}-7\hat{j}+5\hat{k},\vec{b}=\hat{i}+\hat{k}$ and $\vec{c}=\hat{i}+2\hat{j}-3\hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $\vec{r}\times \vec{a}=\vec{c}\times \vec{a}$ and $\vec{r}\cdot \vec{b}=0$, then $|\vec{r}|$ is equal to:

Let $\vec{a},\vec{b}$ and $\vec{c}$ be three non-zero non-coplanar vectors. Let the position vectors of four points $A,B,C$ and $D$ be $\vec{a}-\vec{b}+\vec{c},\lambda \vec{a}-3\vec{b}+4\vec{c}$, $-\vec{a}+2\vec{b}-3\vec{c}$ and $2\vec{a}-4\vec{b}+6\vec{c}$ respectively. If $\vec{AB}$, $\vec{AC}$ and $\vec{AD}$ are coplanar, then $\lambda$ is :

Let $\vec{a}$ and $\vec{b}$ be two vector such that $|\vec{a}|=\sqrt{14},|\vec{b}|=\sqrt{6}$ and $|\vec{a}\times \vec{b}|=\sqrt{48}$. Then ${(\vec{a}\cdot \vec{b})}^{2}$ is equal to _____ .

Let $\vec{a}$and $\vec{b}$ be two vectors. Let $|\vec{a}|=1,|\vec{b}|=4$ and $\vec{a}\cdot \vec{b}=2$. If $\vec{c}=(2\vec{a}\times \vec{b})-3\vec{b}$, then the value of $\vec{b}\cdot \vec{c}$ is

Let $\vec{a}=5\hat{i}-\hat{j}-3\hat{k}$ and $\vec{b}=\hat{i}+3\hat{j}+5\hat{k}$ be two vectors. Then which one of the following statements is TRUE?

Let $\vec{a}=6\hat{i}+9\hat{j}+12\hat{k},\vec{b}=\alpha \hat{i}+11\hat{j}-2\hat{k}$ and $\vec{c}$ be vectors such that $\vec{a}\times \vec{c}=\vec{a}\times \vec{b}$ If $\vec{a}\cdot \vec{c}=-12,$ and $\vec{c}\cdot (\hat{i}-2\hat{j}+\hat{k})=5$ then $\vec{c}\cdot (\hat{i}+\hat{j}+\hat{k})$ is equal to $_______$

Let $\vec{a}=\hat{i}+4\hat{j}+2\hat{k},\vec{b}=3\hat{i}-2\hat{j}+7\hat{k}$ and $\vec{c}=2\hat{i}-\hat{j}+4\hat{k}$. If a vector $\vec{d}$ satisfies $\vec{d}\times \vec{b}=\vec{c}\times \vec{b}$ and $\vec{d}\cdot \vec{a}=24,$ then ${|\vec{d}|}^{2}$ is equal to

Let $\vec{a}=2\hat{i}+3\hat{j}+4\hat{k},\vec{b}=\hat{i}-2\hat{j}-2\hat{k}$ and $\vec{c}=-\hat{i}+4\hat{j}+3\hat{k}.$ If $\vec{d}$ is a vector perpendicular to both $\vec{b}$ and $\vec{c},$ and $\vec{a}\cdot \vec{d}=18,$ then $|\vec{a}\times \vec{d}{|}^{2}$ is equal to

Let $\vec{a}=3\hat{i}+\hat{j}-\hat{k}$ and $\vec{c}=2\hat{i}-3\hat{j}+3\hat{k}.$ If $\vec{b}$ is a vector such that $\vec{a}=\vec{b}\times \vec{c}$ and $|\vec{b}{|}^{2}=50,$ then $|72-{|\vec{b}+\vec{c}|}^{2}|$ is equal to $__________.$

Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}-\hat{k}$. If $\vec{c}$ is a vector such that $\vec{a}\cdot \vec{c}=11,\vec{b}\cdot (\vec{a}\times \vec{c})=27$ and $\vec{b}\cdot \vec{c}=$ $-\sqrt{3}|\vec{b}|$, then $|\vec{a}\times \vec{c}{|}^{2}$ is equal to

Let $\lambda \in \mathbb{Z},\vec{a}=\lambda \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=3\hat{i}-\hat{j}+2\hat{k}$. Let $\vec{c}$ be a vector such that $(\vec{a}+\vec{b}+\vec{c})\times \vec{c}=\vec{0},\vec{a}\cdot \vec{c}=-17$ and $\vec{b}\cdot \vec{c}=-20$. Then ${|\vec{c}\times (\lambda \hat{i}+\hat{j}+\hat{k})|}^{2}$ is equal to

Let $\vec{a}=2\hat{i}+7\hat{j}-\hat{k},\hat{b}=3\hat{i}+5\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}+2\hat{k}$ Let $\vec{d}$ be a vector which is perpendicular to both $\vec{a}$ and $\vec{b}$, and $\vec{c}\cdot \vec{d}=12$. Then$(-\hat{i}+\hat{j}-\hat{k})\cdot (\vec{c}\times \vec{d})$ is equal to

Let $\vec{\alpha }=4\hat{i}+3\hat{j}+5\hat{k}$ and $\vec{\beta }=\hat{i}+2\hat{j}-4\hat{k}$. Let ${\vec{\beta }}_{1}$ be parallel to $\vec{\alpha }$ and ${\vec{\beta }}_{2}$ be perpendicular to $\vec{\alpha }$. If $\vec{\beta }={\vec{\beta }}_{1}+{\vec{\beta }}_{2}$, then the value of $5{\vec{\beta }}_{2}\cdot (\hat{i}+\hat{j}+\hat{k})$ is

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

Let $\vec{u}=\hat{i}-\hat{j}-2\hat{k},\vec{v}=2\hat{i}+\hat{j}-\hat{k}$, $\vec{v}\cdot \vec{w}=2$ and $\vec{v}\times \vec{w}=\vec{u}+\lambda \vec{v}$, then $\vec{u}\cdot \vec{w}$ is equal to

Let $\vec{a}=-\hat{i}-\hat{j}+\hat{k},\vec{a}\cdot \vec{b}=1$ and $\vec{a}\times \vec{b}=\hat{i}-\hat{j}$. Then $\vec{a}-6\vec{b}$ is equal to

Let $\vec{a}$ be a non-zero vector parallel to the line of intersection of the two planes described by $\hat{i}+\hat{j},\hat{i}+\hat{k}$ and $\hat{i}-\hat{j},\hat{j}-\hat{k}$. If $\theta$ is the angle between the vector $\vec{a}$ and the vector $\vec{b}=2\hat{i}-2\hat{j}+\hat{k}$ and $\vec{a}\cdot \vec{b}=6$, then the ordered pair $(\theta ,|\vec{a}\times \vec{b}|)$ is equal to

Let $ABCD$ be a quadrilateral. If $E$ and $F$ are the mid points of the diagonals $AC$ and $BD$ respectively and $(\vec{AB}-\vec{BC})+(\vec{AD}-\vec{DC})=k\vec{FE}$, then $k$ is equal to

Let $N$ be the foot of perpendicular from the point$P(1,-2,3)$ on the line passing through the points $(4,5,8)$ and $(1,-7,5)$. Then the distance of $N$ from the plane $2x-2y+z+5=0$ is

Let $O$ be the origin and the position vector of the point $P$ be $-\hat{i}-2\hat{j}+3k$. If the position vectors of the points $A,B$ and $C$ are $-2\hat{i}+\hat{j}-3k,2\hat{i}+4\hat{j}-2k$ and $-4\hat{i}^+2\hat{j}-k$ respectively, then the projection of the vector $\vec{OP}$ on a vector perpendicular to the vectors $\vec{AB}$ and $\vec{AC}$ is

Let $S$ be the set of all $(\lambda ,\mu )$ for which the vectors $\lambda \hat{i}-\hat{j}+\hat{k},\hat{j}+2\hat{j}+\mu \hat{k}$ and $3\hat{i}-4\hat{j}+5\hat{k}$, where $\lambda -\mu =5$, are coplanar, then $\underset{(\lambda ,\mu )\in S}{\sum }80({\lambda }^{2}+{\mu }^{2})$ is equal to

Let $S$ be the set of all values of $\lambda$, for which the shortest distance between the lines $\frac{x-\lambda }{0}=\frac{y-3}{4}=\frac{z+6}{1}$and $\frac{x+\lambda }{3}=\frac{y}{-4}=\frac{z-6}{0}$ is $13$. Then $8|\underset{\lambda \in S}{\sum }\lambda |$ is equal to

Let ${\lambda }_{1},{\lambda }_{2}$ be the values of $\lambda$ for which the points $(\frac{5}{2},1,\lambda )$ and $(-2,0,1)$ are at equal distance from the plane $2x+3y-6z+7$ If ${\lambda }_{1}>{\lambda }_{2}$ then the distance of the point $({\lambda }_{1}-{\lambda }_{2},{\lambda }_{2},{\lambda }_{1})$ from the line $\frac{x-5}{1}=\frac{y-1}{2}=\frac{z+7}{2}$ is $______$

Let $a,b,c$ be three distinct real numbers, none equal to one. If the vectors $a\hat{i}+\hat{j}+\hat{k},\hat{i}+b\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+c\hat{k}$ are coplanar, then $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is equal to

Let $\vec{a},\vec{b},\vec{c}$ be three vectors such that $|\vec{a}|=\sqrt{31},4|\vec{b}|=|\vec{c}|=2$ and $2(\vec{a}\times \vec{b})=3(\vec{c}\times \vec{a})$. If the angle between $\vec{b}$ and $\vec{c}$ is $\frac{2\pi }{3}$, then ${(\frac{\vec{a}\times \vec{c}}{\vec{a}\cdot \vec{b}})}^{2}$ is equal to _____ .

Let the co-ordinates of one vertex of $\Delta ABC$ be $A(0,2,\alpha )$ and the other two vertices lie on the line $\frac{x+\alpha }{5}=\frac{y-1}{2}=\frac{z+4}{3}$. For $\alpha \in \mathbb{Z}$, if the area of $\Delta ABC$ is $21\mathrm{sq}.\mathrm{units}$ and the line segment $BC$ has length $2\sqrt{21}$ units, then ${\alpha }^{2}$ is equal to _______.

Let the image of the point $P(1,2,3)$ in the plane $2x–y+z=9$ be $Q.$ If the coordinates of the point $R$ are $(6,10,7),$ then the square of the area of the triangle $PQR$ is $_______.$

Let the line $\frac{x}{1}=\frac{6-y}{2}=\frac{z+8}{5}$ intersect the lines $\frac{x-5}{4}=\frac{y-7}{3}=\frac{z+2}{1}$ and $\frac{x+3}{6}=\frac{3-y}{3}=\frac{z-6}{1}$ at the points $A$ and $B$ respectively. Then the distance of the mid-point of the line segment $AB$ from the plane $2x-2y+z=14$ is

Let the line $L$ pass through the point $(0,1,2),$ intersect the line $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and be parallel to the plane $2x+y-3z=4$. Then the distance of the point $P(1,–9,2)$ from the line $L$ is

Let the position vectors of the points $A,B,C$ and $D$ be $5\hat{i}+5\hat{j}+2\lambda \hat{k},\hat{i}+2\hat{j}+3\hat{k},-2\hat{i}+\lambda \hat{j}+4\hat{k}$ and $-\hat{i}+5\hat{j}+6\hat{k}.$ Let the set $S={\lambda \in \mathbb{R}:$ the points $A,B,C$ and $D$ are coplanar$}.$ The $\underset{\lambda \in S}{\sum }(\lambda +2{)}^{2}$ is equal to

Let the shortest distance between the lines $L:\frac{x-5}{-2}=\frac{y-\lambda }{0}=\frac{z+\lambda }{1},\lambda \geq 0$ and ${L}_{1}:x+1=y-1=4-z$ be $2\sqrt{6}$. If $(\alpha ,\beta ,\gamma )$ lies on $L$, then which of the following is NOT possible?

Let the vectors $\vec{{u}_{1}}=\hat{i}+\hat{j}+a\hat{k},\vec{{u}_{2}}=\hat{i}+b\hat{j}+\hat{k},$ and $\vec{{u}_{3}}=c\hat{i}+\hat{j}+\hat{k}$ be coplanar. If the vectors $\vec{{v}_{1}}=(a+b)\hat{i}+c\hat{j}+c\hat{k},\vec{{v}_{2}}=a\hat{i}+(b+c)\hat{j}+a\hat{k}$ and ${\vec{v}}_{3}=b\hat{i}+b\hat{j}+(c+a)\hat{k}$ are also coplanar, then $6(a+b+c)$ is equal to

Let $\vec{a}=\hat{i}+2\hat{j}+\lambda \hat{k}$, $\vec{b}=3\hat{i}-5\hat{j}-\lambda \hat{k}$, $\vec{a}\cdot \vec{c}=7$, $2(\vec{b}\cdot \vec{c})+43=0$, $\vec{a}\times \vec{c}=\vec{b}\times \vec{c}$, then $\vec{a}\cdot \vec{b}$ is equal to

One vertex of a rectangular parallelopiped is at the origin $O$ and the lengths of its edges along $x,y$ and $z$ axes are $3,4$ and $5$ units respectively. Let $P$ be the vertex $(3,4,5).$ Then the shortest distance between the diagonal $OP$ and an edge parallel to $z$ axis, not passing through $O$ or $P$ is

Shortest distance between the lines $\frac{x-1}{2}=\frac{y+8}{-7}=\frac{z-4}{5}\text{ and }\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-6}{-3}\text{ is }$

The area of the quadrilateral $ABCD$ with vertices $A(2,1,1),B(1,2,5),C(-2,-3,5)$ and $D(1,-6,-7)$ is equal to

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 foot of perpendicular of the point $(2,0,5)$ on the line $\frac{x+1}{2}=\frac{y-1}{5}=\frac{z+1}{-1}$ is $(\alpha ,\beta ,\gamma )$. Then. Which of the following is NOT correct?

The line ${l}_{1}$ passes through the point $(2,6,2)$ and is perpendicular to the plane $2x+y-2z=10$. Then the shortest distance between the line ${l}_{1}$ and the line $\frac{x+1}{2}=\frac{y+4}{-3}=\frac{z}{2}$ is:

The shortest distance between the lines $x+1=2y=-12z$ and $x=y+2=6z-6$ is

The shortest distance between the lines $\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-6}{2}$ and $\frac{x-6}{3}=\frac{1-y}{2}=\frac{z+8}{0}$ is equal to ______

The shortest distance between the lines $\frac{x-5}{1}=\frac{y-2}{2}=\frac{z-4}{-3}$ and $\frac{x+3}{1}=\frac{y+5}{4}=\frac{z-1}{-5}$ is

The shortest distance between the lines $\frac{x-4}{4}=\frac{y+2}{5}=\frac{z+3}{3}$ and $\frac{x-1}{3}=\frac{y-3}{4}=\frac{z-4}{2}$ is

The shortest distance between the lines $\frac{x+2}{1}=\frac{y}{-2}=\frac{z-5}{2}$ and $\frac{x-4}{1}=\frac{y-1}{2}=\frac{z+3}{0}$ is

The sum of all values of $\alpha$, for which the points whose position vectors are $\hat{i}-2\hat{j}+3\hat{k},2\hat{i}-3\hat{j}+4\hat{k},(\alpha +1)\hat{i}+2\hat{k}$ and $9\hat{i}+(\alpha -8)\hat{j}+6\hat{k}$ are coplanar, is equal to

The vector $\vec{a}=-\hat{i}+2\hat{j}+\hat{k}$ is rotated through a right angle, passing through the $y$-axis in its way and the resulting vector is $\vec{b}$. Then the projection of $3\vec{a}+\sqrt{2}\vec{b}$ on $\vec{c}=5\hat{i}+4\hat{j}+3\hat{k}$ is