If the vectors $3i + \lambda j + k$ and $2i - j + 8k$ are perpendicular,then $\lambda$ is:

Find the angle between the vectors $\hat{i}-2 \hat{j}+3 \hat{k}$ and $3 \hat{i}-2 \hat{j}+\hat{k}$.

For what values of $x$ is the angle between the vectors $\vec{a} = x\hat{i} - 3\hat{j} - \hat{k}$ and $\vec{b} = 2x\hat{i} + x\hat{j} - \hat{k}$ acute,and the angle between the vector $\vec{b}$ and the $y$-axis obtuse?

Let $\bar{a}, \bar{b}, \bar{c}, \bar{d}$ be vectors such that $\bar{a} \times \bar{b} = 2\hat{i} + 3\hat{j} - \hat{k}$ and $\bar{c} \times \bar{d} = 3\hat{i} + 2\hat{j} + \lambda\hat{k}$. If $\begin{vmatrix} \bar{a} \cdot \bar{c} & \bar{b} \cdot \bar{c} \\ \bar{a} \cdot \bar{d} & \bar{b} \cdot \bar{d} \end{vmatrix} = 0$,then find the value of $\lambda$.

$a, b$ and $c$ are three vectors such that $|a|=1, |b|=2, |c|=3$ and $b, c$ are perpendicular. If the projection of $b$ on $a$ is the same as the projection of $c$ on $a$,then $|a-b+c|$ is equal to

The vector projection of $\overline{PQ}$ on $\overline{AB}$,where $P \equiv (-2, 1, 3)$,$Q \equiv (3, 2, 5)$,$A \equiv (4, -3, 5)$ and $B \equiv (7, -5, -1)$ is