JEE Main Mathematics — Vectors & 3D Geometry previous year questions with solutions.
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?
Shortest distance between the lines $\frac{x-1}{2}=\frac{y+8}{-7}=\frac{z-4}{5}\text{ and }\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-6}{-3}\text{ is }$
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 _____ .
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 _______.
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
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 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:
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
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
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 ________
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 :
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
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
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 _____ .
The shortest distance between the lines $x+1=2y=-12z$ and $x=y+2=6z-6$ is
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 $S$ 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 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 $O$ 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 $\hat{a}$ 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 $\vec{a}=\alpha \hat{i}+2\hat{j}-\hat{k}$ 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 $\vec{a}=\hat{i}+\hat{j}-\hat{k}$ 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 ${l}_{1}$ 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 _____.
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