If $(\vec{a}+3 \vec{b})$ is perpendicular to $(7 \vec{a}-5 \vec{b})$ and $(\vec{a}-4 \vec{b})$ is perpendicular to $(7 \vec{a}-2 \vec{b})$,then the angle between $\vec{a}$ and $\vec{b}$ (in degrees) is $......$

If the non-zero vectors $a$ and $b$ are perpendicular to each other,then the solution of the equation $r \times a = b$ is given by

If $\vec{a} = 2\hat{i} + 3\hat{j} + \hat{k}$,$\vec{b} = \hat{i} - \hat{j} + \hat{k}$,$\vec{c} = \hat{i} + \hat{j} + \hat{k}$ and let $\vec{d}$ be such that $\vec{a} \times \vec{b} = \vec{d} \times \vec{b}$ and $\vec{d} \cdot \vec{c} = 8$,then the value of $\vec{d} \cdot \vec{b}$ is:

The orthogonal projection vector of $\vec{a} = 2\hat{i} + 3\hat{j} + 3\hat{k}$ on $\vec{b} = \hat{i} - 2\hat{j} + \hat{k}$ is

The projection of vector $\vec{a}$ along vector $\vec{b}$ is:

The projection of $\overline{AB}$ on $\overline{CD}$,where $A \equiv (2, -3, 0)$,$B \equiv (1, -4, -2)$,$C \equiv (4, 6, 8)$,and $D \equiv (7, 0, 10)$ is