The position vectors of the points $A, B, C$ are $(2i + j - k)$,$(3i - 2j + k)$,and $(i + 4j - 3k)$ respectively. These points

If the position vectors of $P$ and $Q$ are $(i + 3j - 7k)$ and $(5i - 2j + 4k)$,then $|\overrightarrow{PQ}|$ is

$A$ vector $\vec{a}$ has components $2p$ and $1$ with respect to a rectangular Cartesian system. This system is rotated through a certain angle about the origin in the counter-clockwise sense. If,with respect to the new system,$\vec{a}$ has components $p+1$ and $1$,then:

If the position vectors of the points $A, B, C, D$ are $\vec{a} = \hat{i} + \hat{j} + \hat{k}$,$\vec{b} = 2\hat{i} + 5\hat{j}$,$\vec{c} = 3\hat{i} + 2\hat{j} - 3\hat{k}$,and $\vec{d} = \hat{i} - 6\hat{j} - \hat{k}$,then the angle between the vectors $\overrightarrow{AB}$ and $\overrightarrow{CD}$ is:

Let $a = i$ be a vector which makes an angle of $120^\circ$ with a unit vector $b$. Then the unit vector $(a + b)$ is

If the position vectors of the points $A, B, C, D$ are $2i + 3j + 5k, i + 2j + 3k, -5i + 4j - 2k$ and $i + 10j + 10k$ respectively,then: