If $\vec{A} + \vec{B} + \vec{C} = 0$,then $\vec{A} \times \vec{B}$ is equal to:

If $\vec{A} \times \vec{B} = \vec{B} \times \vec{C} = \vec{C} \times \vec{A}$,then $\vec{A} + \vec{B} + \vec{C}$ is equal to:

$P$ and $Q$ are two non-zero vectors inclined to each other at an angle $\theta$. The component of $Q$ in the direction of $P$ is:

Calculate the values of $a$ and $b$ if vectors $a\hat{i} + b\hat{j} = \hat{n}$ and $(\hat{i} + \hat{j})$ are perpendicular to each other. Note: $\hat{n}$ is a unit vector.

For three vectors $\vec{A} = (-x \hat{i} - 6 \hat{j} - 2 \hat{k})$,$\vec{B} = (-\hat{i} + 4 \hat{j} + 3 \hat{k})$ and $\vec{C} = (-8 \hat{i} - \hat{j} + 3 \hat{k})$,if $\vec{A} \cdot (\vec{B} \times \vec{C}) = 0$,then the value of $x$ is:

Explain the meaning of multiplication of vectors by real numbers with an example.