Vectors Questions with Solutions

If the volume of the parallelepiped whose adjacent edges are $\hat{a}=2\hat{i}+3\hat{j}+4\hat{k}$, $\hat{b}=\hat{i}+\alpha \hat{j}+2\hat{k}$ and $\hat{c}=\hat{i}+2\hat{j}+\alpha \hat{k}$ is 15, then $\alpha$ is equal to

Let $\hat{a}=2\hat{i}+2\hat{j}+\hat{k}$ and $\hat{b}$ be another vector such that $\hat{a}\cdot \hat{b}=14$ and $\hat{a}\times \hat{b}=3\hat{i}+\hat{j}-8\hat{k}$. Then the vector $\hat{b}$ is

If $\hat{a}=\lambda \hat{i}+\hat{j}-2\hat{k}$, $\hat{b}=\hat{i}+\lambda \hat{j}-2\hat{k}$, $\hat{c}=\hat{i}+\hat{j}+\hat{k}$ and $\left[\hat{a}\hat{b}\hat{c}\right]=7$, then the values of $\lambda$ are

If $\left(\hat{a}\times \hat{b}\right)\times \hat{c}=\hat{a}\times \left(\hat{b}\times \hat{c}\right)$, then

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

If $\hat{A}=4\hat{i}+3\hat{j}+\hat{k}$ and $\hat{B}=2\hat{i}-\hat{j}+2\hat{k}$, then the unit vector $\hat{N}$ perpendicular to the vectors $\hat{A}$ and $\hat{B}$, such that $\hat{A}$, $\hat{B}$ and $\hat{N}$ form a right handed system, 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

If $\hat{a}$ and $\hat{b}$ are vectors in space given by $\hat{a}=\frac{\hat{i}-2\hat{j}}{\sqrt{5}}$ and $\hat{b}=\frac{2\hat{i}+\hat{j}+3\hat{k}}{\sqrt{14}}$, then the value of $\left(2\hat{a}+\hat{b}\right)\cdot \left[\left(\hat{a}\times \hat{b}\right)\times \left(\hat{a}-2\hat{b}\right)\right]$ is

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

Let $\hat{a}=\hat{i}+\hat{j}+\hat{k}$, $\hat{b}=\hat{i}-\hat{j}+\hat{k}$ and $\hat{c}=\hat{i}-\hat{j}-\hat{k}$ be three vectors. A vector $\hat{v}$ in the plane of $\hat{a}$ and $\hat{b}$ whose projection on $\frac{\hat{c}}{|\hat{c}|}$ is $\frac{1}{\sqrt{3}}$ is

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

If $\widehat{AC}=2\hat{i}+\hat{j}+\hat{k}$ and $\widehat{BD}=-\hat{i}+3\hat{j}+2\hat{k}$, then the area of the quadrilateral ABCD is

If $\hat{a}$ and $\hat{b}$ are vectors such that $\left|\hat{a}\right|=13$, $\left|\hat{b}\right|=5$ and $\hat{a}\cdot \hat{b}=60$, then the value of $\left|\hat{a}\times \hat{b}\right|$ is

If $\hat{a}$, $\hat{b}$ and $\hat{c}$ are vectors such that $\hat{a}+\hat{b}+\hat{c}=\hat{0}$ and $\left|\hat{a}\right|=7$, $\left|\hat{b}\right|=5$, $\left|\hat{c}\right|=3$, then the angle between the vectors $\hat{b}$ and $\hat{c}$ is

Let $\hat{a}$, $\hat{b}$ and $\hat{c}$ be three vectors having magnitudes 1, 1 and 2 respectively. If $\hat{a}\times \left(\hat{a}\times \hat{c}\right)-\hat{b}=\hat{0}$, then the acute angle between $\hat{a}$ and $\hat{c}$ is

Let $\hat{a}$, $\hat{b}$ and $\hat{c}$ be three vectors such that $\left|\hat{a}\right|=2$, $\left|\hat{b}\right|=3$, $\left|\hat{c}\right|=5$ and $\hat{a}+\hat{b}+\hat{c}=\hat{0}$. The value of $\hat{a}\cdot \hat{b}+\hat{b}\cdot \hat{c}+\hat{c}\cdot \hat{a}$ is

If $\hat{a}=\hat{i}+2\hat{j}-3\hat{k}$ and $\hat{b}=3\hat{i}-\hat{j}+2\hat{k}$, then the angle between $\left(\hat{a}+\hat{b}\right)$ and $\left(\hat{a}-\hat{b}\right)$ is

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