Vectors Questions with Solutions

Let $\hat{a},\hat{b},\hat{c}$ be three non-zero vectors, no two of which are collinear. If $\hat{a}+\hat{b}$ is collinear with $\hat{c}$ and $\hat{b}+\hat{c}$ is collinear with $\hat{a}$, then $\hat{a}+\hat{b}+\hat{c}$ is equal to

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

If $\hat{a}$, $\hat{b}$ and $\hat{c}$ are unit vectors, then ${\left|\hat{a}-\hat{b}\right|}^2+{\left|\hat{b}-\hat{c}\right|}^2+{\left|\hat{c}-\hat{a}\right|}^2$ does not exceed

The vector $\hat{B}=3\hat{i}+4\hat{k}$ is to be written as the sum of a vector $\widehat{B_1}$ parallel to $\hat{A}=\hat{i}+\hat{j}$ and a vector $\widehat{B_2}$ perpendicular to $\hat{A}$. Then $\widehat{B_1}$ is:

If $\hat{a},\hat{b},\hat{c}$ are non-coplanar unit vectors such that $\hat{a}\times \left(\hat{b}\times \hat{c}\right)=\frac{\hat{b}+\hat{c}}{\sqrt{2}}$, then the angle between $\hat{a}$ and $\hat{b}$ is:

The value of $\lambda$ for which the volume of the parallelepiped formed by the vectors $\hat{i}+\lambda \hat{j}+\hat{k}$, $\hat{j}+\lambda \hat{k}$ and $\lambda \hat{i}+\hat{k}$ is minimum is

The value of $\lambda$ such that the four points whose position vectors are $3\hat{i}-2\hat{j}+\lambda \hat{k}$, $6\hat{i}+3\hat{j}+\hat{k}$, $5\hat{i}+7\hat{j}+3\hat{k}$ and $2\hat{i}+2\hat{j}+6\hat{k}$ are coplanar is

Let $\hat{A}=2\hat{i}+\hat{j}-2\hat{k}$ and $\hat{B}=\hat{i}+\hat{j}$. If $\hat{C}$ is a vector such that $\hat{A}\cdot \hat{C}=\left|\hat{C}\right|$, $\left|\hat{C}-\hat{A}\right|=2\sqrt{2}$ and the angle between $\hat{A}\times \hat{B}$ and $\hat{C}$ is ${30}^{\circ }$, then $\left|\left(\hat{A}\times \hat{B}\right)\times \hat{C}\right|$ is equal to

A rigid body is rotating at the rate of $3$ radians per second about an axis $AB$, where $A\left(1,-2,1\right)$ and $B\left(3,-4,2\right)$. The velocity of the point $P\left(5,-1,-1\right)$ of the body is

If $\hat{A}+\hat{B}+\hat{C}=\hat{0}$, $\left|\hat{A}\right|=3$, $\left|\hat{B}\right|=5$ and $\left|\hat{C}\right|=7$, then the angle between $\hat{A}$ and $\hat{B}$ is

Forces of magnitude 5, 3, 1 units act in the directions $6\hat{i}+2\hat{j}+3\hat{k}$, $3\hat{i}-2\hat{j}+6\hat{k}$, $2\hat{i}-3\hat{j}-6\hat{k}$ respectively on a particle which is displaced from the point (2, −1, −3) to (5, −1, 1). The total work done by the force is

If $\hat{a},\hat{b},\hat{c}$ are three non-zero vectors with no two of which are collinear, $\hat{a}+2\hat{b}$ is collinear with $\hat{c}$ and $\hat{b}+3\hat{c}$ is collinear with $\hat{a}$, then $\left|\hat{a}+2\hat{b}+6\hat{c}\right|$ will be equal to

Vertices of the vectors $\hat{i}-2\hat{j}+2\hat{k}$, $2\hat{i}+\hat{j}-\hat{k}$ and $3\hat{i}-\hat{j}+2\hat{k}$ form a triangle. This triangle is

Let $\hat{a}=2\hat{i}+\hat{j}+2\hat{k}$, $\hat{b}=\hat{i}-\hat{j}+2\hat{k}$ and $\hat{c}=\hat{i}+\hat{j}-2\hat{k}$ be three vectors. Then a vector in the plane of $\hat{a}$ and $\hat{c}$ whose projection on $\hat{b}$ is of magnitude $\frac{1}{\sqrt{6}}$ is

If the position vectors of A and B relative to O are $\hat{i}-4\hat{j}+3\hat{k}$ and $-\hat{i}+2\hat{j}-\hat{k}$ respectively, then the median through O of $△OAB$ is

The area of the triangle whose vertices have position vectors $3\hat{i}+\hat{j}$, $5\hat{i}+2\hat{j}+\hat{k}$ and $\hat{i}-2\hat{j}+3\hat{k}$ is

If the vectors $a\hat{i}+\hat{j}+\hat{k}$, $\hat{i}+b\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+c\hat{k}$ $\left(a,b,c\ne 1\right)$ are coplanar, then $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=$

Let $\hat{a}=\hat{i}+\hat{j}$ and $\hat{b}=2\hat{i}-\hat{k}$. The point of intersection of the lines $\hat{r}\times \hat{a}=\hat{b}\times \hat{a}$ and $\hat{r}\times \hat{b}=\hat{a}\times \hat{b}$ is

The angle between $\hat{a}$ and $\hat{b}$ is ${120}^{\circ }$. If $\left|\hat{b}\right|=2\left|\hat{a}\right|$ and the vectors $\hat{a}+x\hat{b}$ and $\hat{a}-\hat{b}$ are at right angles, then $x=$

Let $a,b,c$ 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