Vector Algebra — JEE Main PYQs

Let $\vec{a},\vec{b}$ and $\vec{c}$ be three vectors such that $|\vec{a}|=\sqrt{3},|\vec{b}|=5,\vec{b}\cdot \vec{c}=10$ and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi }{3}$. If $\vec{a}$ is perpendicular to the vector $\vec{b}\times \vec{c}$, then …

If the vectors, $\vec{p}=(a+1)\hat{i}+a\hat{j}+a\hat{k}$, $\vec{q}=a\hat{i}+(a+1)\hat{j}+a\hat{k}$ and $\overrightarrow{\mathrm{r}}=\mathrm{a}\hat{\mathrm{i}}+\mathrm{a}\hat{\mathrm{j}}+(\mathrm{a}+1)\hat{\mathrm{k}}$ $(\mathrm{a}\in \mathrm{R})$ are coplanar and …

Let the vectors $\vec{a},\vec{b},\vec{c}$ be such that $|\vec{a}|=2,|\vec{b}|=4$ and $|\vec{c}|=4$. If the projection of $\vec{b}$ on $\vec{a}$ is equal to the projection of $\vec{c}$ on $\vec{a}$ and $\vec{b}$ is perpendicular to $\vec{c}$, then the value of …

Let $\mathrm{a},\mathrm{b},\mathrm{c}\in \mathrm{R}$ be such that ${\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2=1$. If $a\cos \theta =b\cos \left(\theta +\frac{2\pi }{3}\right)=c\cos \left(\theta +\frac{4\pi }{3}\right)$, where $\theta =\frac{\pi }{9}$, then the angle be …

If $\vec{x}$ and $\vec{y}$ be two non-zero vectors such that $|\vec{x}+\vec{y}|=|\vec{x}|$ and $2\vec{x}+\lambda \vec{y}$ is perpendicular to $\vec{y}$, then the value of $\lambda$ is

If the volume of a parallelepiped, whose coterminus edges are given by the vectors $\vec{a}=\hat{i}+\hat{j}+n\hat{k},\vec{b}=2\hat{i}+4\hat{j}-n\hat{k}$ and $\vec{c}=\hat{i}+n\hat{j}+3\hat{k}(n\ge 0)$, is 158 cu.units, then:

If $\vec{a}$ and $\vec{b}$ are unit vectors, then the greatest value of $\sqrt{3}|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|$ is ______.

If $\vec{a}=2\hat{i}+\hat{j}+2\hat{k}$, then the value of $|\hat{i}\times (\vec{a}\times \hat{i}){|}^2+|\hat{j}\times (\vec{a}\times \hat{j}){|}^2+|\hat{k}\times (\vec{a}\times \hat{k}){|}^2$ is equal to:

The magnitude of the projection of the vector $2\hat{i}+3\hat{j}+\hat{k}$ on the vector perpendicular to the plane containing the vectors $\hat{i}+\hat{j}+\hat{k}$ and $\hat{i}+2\hat{j}+3\hat{k}$, 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 $|\vec{a}\times \vec{b}|=r$ is possible if:

Let $\vec{\alpha }=3\hat{i}+\hat{j}$ and $\vec{\beta }=2\hat{i}-\hat{j}+3\hat{k}$. If $\vec{\beta }=\overrightarrow{{\beta }_1}-\overrightarrow{{\beta }_2}$, where $\overrightarrow{{\beta }_1}$ is parallel to $\vec{\alpha }$ and $\overrightarrow{{\beta }_2}$ is perpendicul …

Let $\vec{a}=2\hat{i}+{\lambda }_1\hat{j}+3\hat{k},\vec{b}=4\hat{i}+\left(3-{\lambda }_2\right)\hat{j}+6\hat{k}$ and $\vec{c}=3\hat{i}+6\hat{j}+\left({\lambda }_3-1\right)\hat{k}$ be three vectors such that $\vec{b}=2\vec{a}$ and $\vec{a}$ is perpendicular to $\vec{c}$. T …

Let $\sqrt{3}\hat{i}+\hat{j}$, $\hat{i}+\sqrt{3}\hat{j}$ and $\beta \hat{i}+(1-\beta )\hat{j}$ respectively be the position vectors of the points $A$, $B$ and $C$ with respect to the origin $O$. If the distance of $C$ from the bisector of the acute angle between OA …

Let $\vec{a}=3\hat{i}+2\hat{j}+2\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}-2\hat{k}$ be two vectors. If a vector perpendicular to both the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$ has the magnitude 12 then one such vector is:

Let $\vec{a}=\hat{i}+2\hat{j}+4\hat{k},\vec{b}=\hat{i}+\lambda \hat{j}+4\hat{k}$ and $\vec{c}=2\hat{i}+4\hat{j}+\left({\lambda }^2-1\right)\hat{k}$ be coplanar vectors. Then the non - zero vector $\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{c}}$ is:

The sum of the distinct real values of $\mu$, for which the vectors, $\mu \hat{i}+\hat{j}+\hat{k},\hat{i}+\mu \hat{j}+\hat{k},\hat{i}+\hat{j}+\mu \hat{k}$ are co-planar, is:

Let $\alpha \in \mathbb{R}$ and the three vectors $\vec{a}=\alpha \hat{i}+\hat{j}+3\hat{k}$, $\vec{b}=2\hat{i}+\hat{j}-\alpha \hat{k}$ and $\vec{c}=\alpha \hat{i}-2\hat{j}+3\hat{k}$. Then the set …

Let $\vec{a},\vec{b}$, and $\vec{c}$ be three unit vectors, out of which vectors $\vec{b}$ and $\vec{c}$ are non-parallel. If $\alpha$ and $\beta$ are the angles which vector $\vec{a}$ makes with vectors $\vec{b}$ and $\vec{c}$ respectively and …

If a unit vector $\vec{r}$ makes angles $\frac{\pi }{3}$ with $\hat{i}$, $\frac{\pi }{4}$ with $\hat{j}$ and $\theta \in (0,\pi )$ with $\hat{k}$, then a value of $\theta$ is

Let $\vec{a}=\hat{i}+\hat{j}+\sqrt{2}\hat{k},\vec{b}=b_1\hat{i}+b_2\hat{j}+\sqrt{2}\hat{k}$ and $\vec{c}=5\hat{i}+\hat{j}+\sqrt{2}\hat{k}$ be three vectors such that the projection vector of $\vec{b}$ on $\vec{a}$ is $\vec{a}$. If $\vec{a}+\vec{b}$ is perpendicular to …

Let $\vec{a}=\hat{i}-\hat{j},\vec{b}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that $\vec{a}\times \vec{c}+\vec{b}=0$ and $\vec{a}\cdot \vec{c}=4$, then $|\vec{c}{|}^2$ is equal to:

Let $A(3,0,-1),B(2,10,6)$ and $C(1,2,1)$ be the vertices of a triangle and $M$ be the midpoint of AC. If G divides BM in the ratio, $2:1$, then $\cos (\angle \mathrm{GOA})$ (O being the origin) is equal to:

Let $\overrightarrow{\mathrm{u}}$ be a vector coplanar with the vectors $\overrightarrow{\mathrm{a}}=2\hat{i}+3\hat{j}-\hat{k}$ and $\overrightarrow{\mathrm{b}}=\hat{j}+\hat{k}$. If $\overrightarrow{\mathrm{u}}$ is perpendicular to $\vec{a}$ and $\vec{u}\cdot \vec{b}=24$, then …

Let $\overrightarrow{\mathrm{a}}=2\hat{i}+\hat{j}-2\hat{k}$ and $\overrightarrow{\mathrm{b}}=\hat{i}+\hat{j}$. Let $\vec{c}$ be a vector such that $|\vec{c}-\vec{a}|=3$, $|(\vec{a}\times \vec{b})\times \vec{c}|=3$ and the angle between …

Let $\vec{a},\vec{b}$ and $\vec{c}$ be three unit vectors such that $\vec{a}\times (\vec{b}\times \vec{c})=\frac{\sqrt{3}}{2}(\vec{b}+\vec{c})$. If $\vec{b}$ is not parallel to $\vec{c}$, then the angle between $\vec{a}$ and $\vec{b}$ is:

Let $\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}$ be three non-zero vectors such that no two of them are collinear and …

If $[\vec{a}\times \vec{b}\vec{b}\times \vec{c}\vec{c}\times \vec{a}]=\lambda [\vec{a}\vec{b}\vec{c}{]}^2$, then $\lambda$ is equal to

If the vectors $\overrightarrow{\mathrm{AB}}=3\hat{\mathrm{i}}+4\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{AC}}=5\hat{\mathrm{i}}-2\hat{\mathrm{j}}+4\hat{\mathrm{k}}$ are the sides of a triangle ABC, then the length of the median through A is

Let $\hat{a}$ and $\hat{b}$ be two unit vectors. If the vectors $\vec{c}=\hat{a}+2\hat{b}$ and $\vec{d}=5\hat{a}-4\hat{b}$ are perpendicular to each other, then the angle between $\hat{a}$ and $\hat{b}$ is

Let ABCD be a parallelogram such that $\overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{q}},\overrightarrow{\mathrm{AD}}=\overrightarrow{\mathrm{p}}$ and $\angle \mathrm{BAD}$ be an acute angle. If $\overrightarrow{\mathrm{r}}$ is the vector that coincides with the altitude …

If $\vec{a}=\frac{1}{\sqrt{10}}(3\hat{i}+\hat{k})$ and $\vec{b}=\frac{1}{7}(2\hat{i}+3\hat{j}-6\hat{k})$, then the value of $(2\vec{a}-\vec{b})\cdot [(\vec{a}\times \vec{b})\times (\vec{a}+2\vec{b})]$ is

The vector $\vec{a}$ and $\vec{b}$ are not perpendicular and $\vec{c}$ and $\vec{d}$ are two vectors satisfying: $\vec{b}\times \vec{c}=\vec{b}\times \vec{d}$ and $\vec{a}\cdot \vec{d}=0$. Then the vector $\vec{d}$ is equal to

Let $\vec{a}=\hat{j}-\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$. Then vector $\vec{b}$ satisfying $\vec{a}\times \vec{b}+\vec{c}=\overrightarrow{0}$ and $\vec{a}\cdot \vec{b}=3$ is

If the vectors $\vec{a}=\hat{i}-\hat{j}+2\hat{k},\vec{b}=2\hat{i}+4\hat{j}+\hat{k}$ and $\vec{c}=\lambda \hat{i}+\hat{j}+\mu \hat{k}$ are mutually orthogonal, then $(\lambda ,\mu )=$

If $\vec{u},\vec{v},\vec{w}$ are non-coplanar vectors and $p,q$ are real numbers, then the equality …

The vector $\vec{a}=\alpha \hat{i}+2\hat{j}+\beta \hat{k}$ lies in the plane of the vectors $\vec{b}=\hat{i}+\hat{j}$ and $\vec{c}=\hat{j}+\hat{k}$ and bisects the angle between $\vec{b}$ and $\vec{c}$. Then which one of the following gives possible values of $\alpha$ and …

The non-zero vectors $\vec{a},\vec{b}$ and $\vec{c}$ are related by $\vec{a}=8\vec{b}$ and $\vec{c}=-7\vec{b}$. Then the angle between $\vec{a}$ and $\vec{c}$ is

The resultant of two forces $PN$ and $3N$ is a force of $7N$. If the direction of $3N$ force were reversed, the resultant would be $\sqrt{19}\mathrm{N}$. The value of $P$ is

If $\hat{u}$ and $\hat{v}$ are unit vectors and $\theta$ is the acute angle between them, then $2\hat{u}\times 3\hat{v}$ is a unit vector for

Let $\bar{a}=\hat{i}+\hat{j}+\hat{k}$, $\bar{b}=\hat{i}-\hat{j}+2\hat{k}$ and $\bar{c}=x\hat{i}+(x-2)\hat{j}-\hat{k}$. If the vector $\bar{c}$ lies in the plane of $\bar{a}$ and $\bar{b}$, then $x$ equals

ABC is a triangle, right angled at A. The resultant of the forces acting along $\overrightarrow{\mathrm{AB}}$, $\overrightarrow{\mathrm{AC}}$ with magnitudes $\frac{1}{AB}$ and $\frac{1}{AC}$ respectively is the force along $\overrightarrow{AD}$, where $D$ is the foot …

If $(\overline{\mathrm{a}}\times \overline{\mathrm{b}})\times \overline{\mathrm{c}}=\overline{\mathrm{a}}\times (\overline{\mathrm{b}}\times \overline{\mathrm{c}})$, where $\overline{\mathrm{a}},\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ are any three vectors su …

A particle has two velocities of equal magnitude inclined to each other at an angle $\theta$. If one of them is halved, the angle between the other and the original resultant velocity is bisected by the new resultant. Then $\theta$ is

The values of $a$, for which the points $A,B,C$ with position vectors $2\hat{i}-\hat{j}+\hat{k},\hat{i}-3\hat{j}-5\hat{k}$ and $a\hat{i}-3\hat{j}+\hat{k}$ respectively are the vertices of a right-angled triangle with $\mathrm{C}=\frac{\pi }{2}$ are

If $C$ is the mid point of $AB$ and $P$ is any point outside $AB$, then

For any vector $\vec{a}$, the value of $(\vec{a}\times \hat{i}{)}^2+(\vec{a}\times \hat{j}{)}^2+(\vec{a}\times \hat{k}{)}^2$ is equal to

Let $a,b$ and $c$ be distinct non-negative numbers. If the vectors $a\hat{i}+a\hat{j}+c\hat{k},\hat{i}+\hat{k}$ and $c\hat{i}+c\hat{j}+b\hat{k}$ lie in a plane, then $c$ is

If $\vec{a},\vec{b},\vec{c}$ are non-coplanar vectors and $\lambda$ is a real number then
$\left[\lambda (\vec{a}+\vec{b}){\lambda }^2\vec{b}\lambda \vec{c}\right]=[\vec{a}\vec{b}+\vec{c}\vec{b}]$ for

Let $\vec{a}=\hat{i}-\hat{k},\vec{b}=x\hat{i}+\hat{j}+(1-x)\hat{k}$ and $\vec{c}=y\hat{i}+x\hat{j}+(1+x-y)\hat{k}$. Then $[\vec{a},\vec{b},\vec{c}]$ depends on

The resultant R of two forces acting on a particle is at right angles to one of them and its magnitude is one third of the other force. The ratio of larger force to smaller one is