Vector $\vec{A}$ of magnitude $5 \sqrt{3}$ units,another vector $\vec{B}$ of magnitude $10$ units are inclined to each other at an angle of $30^{\circ}$. The magnitude of the vector product of the two vectors is $\left[\sin 30^{\circ}=\frac{1}{2}\right]$

For what value of $x$ will the two vectors $\vec{A} = 2\hat{i} + 2\hat{j} - x\hat{k}$ and $\vec{B} = 2\hat{i} - \hat{j} - 3\hat{k}$ be perpendicular to each other?

Explain the cross product of two vectors.

Three vectors $\vec a$,$\vec b$,and $\vec c$ satisfy the relations $\vec a \cdot \vec b = 0$ and $\vec a \cdot \vec c = 0$. The vector $\vec a$ is parallel to:

The angle between the vectors $(\hat{i} + \hat{j})$ and $(\hat{j} + \hat{k})$ is ....... $^\circ$.

Three particles $P, Q$ and $R$ are moving along the vectors $\vec{A}=\hat{i}+\hat{j}, \vec{B}=\hat{j}+\hat{k}$ and $\vec{C}=-\hat{i}+\hat{j}$ respectively. They strike a point and start to move in different directions. Now,particle $P$ is moving normal to the plane containing vectors $\vec{A}$ and $\vec{B}$. Similarly,particle $Q$ is moving normal to the plane containing vectors $\vec{A}$ and $\vec{C}$. The angle between the directions of motion of $P$ and $Q$ is $\cos^{-1}\left(\frac{1}{\sqrt{x}}\right)$. Then the value of $x$ is ...... .