Let $p=\hat{i}+2\hat{j}-\hat{k}$ and $q=2\hat{i}-\hat{j}+\hat{k}$. If $a$ and $b$ are two vectors such that $p=a-2b$ and $q=2a+b$, then the angle between $a$ and $b$ is

If vectors $\vec{a}$ and $\vec{b}$ are not perpendicular,and $\vec{c}$ and $\vec{d}$ are two vectors satisfying $\vec{b} \times \vec{c} = \vec{b} \times \vec{d}$ and $\vec{a} \cdot \vec{d} = 0$,then the vector $\vec{d}$ is equal to:

Let $\overrightarrow{OA}$ and $\overrightarrow{OB}$ be two sides of a triangle. The median $\overrightarrow{AM}$ is perpendicular to the angle bisector $\overrightarrow{OL}$ and $|\overrightarrow{AM}|:|\overrightarrow{OL}|=1:2$. The angle between $\overrightarrow{OA}$ and $\overrightarrow{OB}$ is

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:

If $A, B, C, D$ are the points $(2, 3, -1), (3, 5, -3), (1, 2, 3), (3, 5, 7)$ respectively,then the angle between $AB$ and $CD$ is

If the angle between the vectors $2 \alpha^2 \hat{i} + 4 \alpha \hat{j} + \hat{k}$ and $7 \hat{i} - 2 \hat{j} + \alpha \hat{k}$ is obtuse, then