If two vectors $\vec{P}=\hat{i}+2 m \hat{j}+m \hat{k}$ and $\vec{Q}=4 \hat{i}-2 \hat{j}+ mk$ are perpendicular to each other. Then, the value of $m$ will be :

What will be the projection of vector $A=\hat{i}+\hat{j}+\hat{k}$ on vector $\vec{B}=\hat{i}+\hat{j}$.

Vector product of two vectors $2\hat i\, + \,\hat j\,$ and $\hat i\, + \,2\hat j\,$ is

Given $\left| {{\vec A_1}} \right| = 2,\,\left| {{\vec A_2}} \right| = 3$ and $\left| {{{\vec A}_1} + {{\vec A}_2}} \right| = 3$.  Find the value or $\left| {\left( {{{\vec A}_1} + 2{{\vec A}_2}} \right) \times \left( {3{{\vec A}_1} - 4{{\vec A}_2}} \right)} \right|$

Define the scalar product of two vectors.

The angle between the vectors $\overrightarrow A $ and $\overrightarrow B $ is $\theta .$ The value of the triple product $\overrightarrow A \,.\,(\overrightarrow B \times \overrightarrow A \,)$ is