The angle between force $\vec{F}=3 \hat{i}+4 \hat{j}-5 \hat{k}$ and displacement $\vec{d}=5 \hat{i}+4 \hat{j}+3 \hat{k}$ is

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 $\left[\sin 30^{\circ}=\frac{1}{2}\right]$

Explain the kinds of multiplication operations for vectors.

If $\vec{A} \times \vec{B} = \vec{0}$ and $\vec{B} \times \vec{C} = \vec{0}$,what is the angle between $\vec{A}$ and $\vec{C}$?

What is the angle between $\vec{A}$ and $\vec{B}$ if $\vec{A}$ and $\vec{B}$ are the adjacent sides of a parallelogram drawn from a common point and the area of the parallelogram is $\frac{AB}{2}$?

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 ...... .