Vectors Questions with Solutions

If $\hat{a}=\frac{\hat{i}-2\hat{j}}{\sqrt{5}}$ 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

The vectors $\hat{a},\hat{b},\hat{c}$ are equal in length and, taken pairwise, make equal angles. If $\hat{a}=\hat{i}+\hat{j}$, $\hat{b}=\hat{j}+\hat{k}$ and $\hat{c}$ makes an obtuse angle with $\hat{i}$, then $\hat{c}$ is equal to

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 $\hat{a}$ 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 $\hat{a}=\hat{i}+\hat{j}+\hat{k}$, $\hat{b}=\hat{i}-\hat{j}+\hat{k}$ 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

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

A force of 78 grams (weight) acts at the point (2, 3, 5), the direction ratios of its line of action being 2, 2, 1. The magnitude of its moment about the line joining the origin to the point (12, 3, 4) is