Write the value of $\hat{i}·(\hat{j}\times \hat{k})+\hat{j}·(\hat{i}\times \hat{k})+\hat{k}·(\hat{i}\times \hat{j})$.

Find the volume of the parallelepiped (in cubic units) whose edges are represented by the vectors $\overrightarrow{a}=(2\hat{i}-3\hat{j}+4\hat{k}),\overrightarrow{b}=(\hat{i}+2\hat{j}-\hat{k})$ and $\overrightarrow{c}=(3\hat{i}-2\hat{j}+2\hat{k})$.

If $\overrightarrow{a}=(-2\hat{i}-2\hat{j}+4\hat{k}),\overrightarrow{b}=(-2\hat{j}+4\hat{j}-2\hat{k})$ and $\overrightarrow{c}=(4\hat{i}-2\hat{j}-2\hat{k})$, then prove that $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are coplanar.

If $\overrightarrow{a}=(2\hat{i}+6\hat{j}+27\hat{k})$ and $\overrightarrow{b}=(\hat{i}+\lambda \hat{j}+\mu \hat{k})$ are such that $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{0}$, then find the value of $\lambda +\mu$.

If $\frac{\mathrm{\pi }}{k}$ is the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ and $|\overrightarrow{a}·\overrightarrow{b}|=|\overrightarrow{a}\times \overrightarrow{b}|$, then what is the value of $\frac{k}{2}$?

When does $|\overrightarrow{a}+\overrightarrow{b}|=\left|\overrightarrow{a}\right|+\left|\overrightarrow{b}\right|$ hold?

Find the direction cosines of a vector which is equally inclined to the $x-$ axis, $y-$ axis and $z-$ axis.

If $P(1, 5, 4)$ and $Q(4, 1,-2)$ be the position vectors of two points $P$ and $Q$, find the direction ratios of $\overline{PQ}$.

Find the direction cosines of the vector $\overrightarrow{a}=(\hat{i}+2\hat{j}+3\hat{k})$ .