Vectors Questions with Solutions

A vector of magnitude 5 units makes equal angles with each of the three mutually perpendicular coordinate axes. The sum of the magnitudes of its projections on the three axes is

If the vectors $2\hat{i}-3\hat{j}$, $\hat{i}+\hat{j}-\hat{k}$ and $3\hat{i}-\hat{k}$ form three conterminous edges of a parallelepiped, then the volume of the parallelepiped is

Forces $3\hat{i}+2\hat{j}+5\hat{k}$ and $2\hat{i}+\hat{j}-3\hat{k}$ act on a particle and displace it from the point $2\hat{i}-\hat{j}-3\hat{k}$ to the point $4\hat{i}-3\hat{j}+7\hat{k}$. Then the work done by the forces is

If the three vectors $2\hat{i}-\hat{j}+\hat{k}$, $\hat{i}+2\hat{j}-3\hat{k}$ and $3\hat{i}+\lambda \hat{j}+5\hat{k}$ are coplanar, then $\lambda$ is

If $\hat{a}$, $\hat{b}$, $\hat{c}$ are unit vectors such that $\hat{a}+\hat{b}+\hat{c}=\hat{0}$, then the value of $\hat{a}\cdot \hat{b}+\hat{b}\cdot \hat{c}+\hat{c}\cdot \hat{a}$ is

If $\hat{a}$, $\hat{b}$, $\hat{c}$ are coplanar vectors, then the value of $\left[2\hat{a}-\hat{b}2\hat{b}-\hat{c}2\hat{c}-\hat{a}\right]$ is

If $\left|\hat{a}\times \hat{b}\right|=\left|\hat{a}\cdot \hat{b}\right|$, then the angle between $\hat{a}$ and $\hat{b}$ is

$ABCD$ is a parallelogram with diagonals $AC$ and $BD$. Then $\widehat{AC}-\widehat{BD}$ is equal to

The position vectors of the points $A,B,C,D$ are $\hat{i}+\hat{j}+\hat{k}$, $2\hat{i}+5\hat{j}$, $3\hat{i}+2\hat{j}-3\hat{k}$ and $\hat{i}-6\hat{j}-\hat{k}$ respectively. The angle between $\widehat{AB}$ and $\widehat{CD}$ is

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

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

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

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

Constant forces $\hat{P}=2\hat{i}-5\hat{j}+6\hat{k}$ and $\hat{Q}=-\hat{i}+2\hat{j}-\hat{k}$ act on a particle. The work done when the particle is displaced from A whose position vector is $4\hat{i}-3\hat{j}-2\hat{k}$, to B whose position vector is $6\hat{i}+\hat{j}-3\hat{k}$, is:

For the vectors $\hat{a}=-4\hat{i}+2\hat{j}$, $\hat{b}=2\hat{i}+\hat{j}$ and $\hat{c}=2\hat{i}+3\hat{j}$, if $\hat{c}=m\hat{a}+n\hat{b}$ then the value of m + n is

The sum of two vectors $\hat{a}$ and $\hat{b}$ is a vector $\hat{c}$ such that $\left|\hat{a}\right|=\left|\hat{b}\right|=\left|\hat{c}\right|=2$. Then, the magnitude of $\hat{a}-\hat{b}$ is equal to:

If a vector $\hat{a}$ makes an equal angle with the coordinate axes and has magnitude 3, then the angle between $\hat{a}$ and each of the three coordinate axes is

Let $\hat{a}$ and $\hat{b}$ be two vectors. Which of the following pairs of vectors are not perpendicular to each other?

If $\hat{a}$, $\hat{b}$ and $\hat{c}$ are the position vectors of the vertices A, B, C of a triangle ABC, then the area of the triangle ABC is