$p=2 \hat{i}-3 \hat{j}+\hat{k}, q=\hat{i}+\hat{j}-\hat{k}$. If the vectors $a$ and $b$ are the orthogonal projections of $p$ on $q$ and $q$ on $p$ respectively, then $\frac{a \times b}{a \cdot b}=$

If $\bar{a}$ and $\bar{b}$ are two unit vectors such that $\bar{a}+2 \bar{b}$ and $5 \bar{a}-4 \bar{b}$ are perpendicular to each other,then the angle between $\bar{a}$ and $\bar{b}$ is

Two adjacent sides of a parallelogram $ABCD$ are given by $\vec{AB} = 2\hat{i} + 10\hat{j} + 11\hat{k}$ and $\vec{AD} = -\hat{i} + 2\hat{j} + 2\hat{k}$. The side $\vec{AD}$ is rotated by an acute angle $\alpha$ in the plane of the parallelogram so that $\vec{AD}$ becomes $\vec{AD'}$. If $\vec{AD'}$ makes a right angle with the side $\vec{AB}$, then the cosine of the angle $\alpha$ is given by

If $\bar{a}$ and $\bar{b}$ are mutually perpendicular unit vectors,then $(3\bar{a}+2\bar{b}) \cdot (5\bar{a}-6\bar{b}) = $

Let $\bar{a}, \bar{b}, \bar{c}$ be three vectors such that $|\bar{a}|=1, |\bar{c}|=1, |\bar{b}|=4$,and $|\bar{b} \times \bar{c}|=\sqrt{15}$. If $\lambda \bar{a}=\bar{b}-2 \bar{c}$,then the value of $\lambda$ is

If $\bar{a}$ and $\bar{b}$ are unit vectors and $\theta$ is the angle between them,then $\tan(\theta/2) =$