Vectors Questions with Solutions

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

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 values of the non-zero scalars $\alpha$ and $\beta$ such that, for all vectors $\hat{a}$ and $\hat{b}$, $\alpha \left(2\hat{a}-\hat{b}\right)+\beta \left(\hat{a}+2\hat{b}\right)=8\hat{b}-\hat{a}$, are

If $\left|\hat{F}\right|=40$ N, $\left|\hat{D}\right|=3$ m and $\theta ={60}^{\circ }$ is the angle between them, then the work done by $\hat{F}$ acting through the displacement $\hat{D}$ is

A man starts at the origin O and walks a distance of 3 units in the north-east direction and then walks a distance of 4 units in the north-west direction to reach the point P. Then $\widehat{OP}$ is equal to

How much work does it take to slide a crate for a distance of 25 m along a loading dock by pulling on it with a 180 N force directed at an angle of ${45}^{\circ }$ to the dock?

The length of the projection of $\hat{a}=2\hat{i}+3\hat{j}+\hat{k}$ on $\hat{b}=-2\hat{i}+\hat{j}+2\hat{k}$ is equal to:

If $\hat{a}$ and $\hat{b}$ are two vectors such that $\left|\hat{a}\right|=3$, $\left|\hat{b}\right|=4$ and $\left|\hat{a}+\hat{b}\right|=1$, then the value of $\left|\hat{a}-\hat{b}\right|$ is

If $\hat{a}=\hat{i}+\hat{j}+\hat{k}$, $\hat{b}=2\hat{i}-\hat{j}+3\hat{k}$ and $\hat{c}=\hat{i}-2\hat{j}+\hat{k}$, then a vector of magnitude $\sqrt{22}$ which is parallel to $2\hat{a}-\hat{b}+3\hat{c}$ is