$(r \cdot i)^2 + (r \cdot j)^2 + (r \cdot k)^2 = $

$A$ 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:

Find the position vector of a point $R$ which divides the line joining two points $P$ and $Q$ whose position vectors are $\hat{i}+2 \hat{j}-\hat{k}$ and $-\hat{i}+\hat{j}+\hat{k}$ respectively,in the ratio $2: 1$ internally.

In $\Delta ABC$,if $A(2, 4, -1)$,$B(4, 5, 1)$,and $C(3, 6, -3)$,then $\Delta ABC$ is a ... triangle.

If $P$ and $Q$ are the midpoints of the sides $BC$ and $CD$ of the parallelogram $ABCD$,then $\overrightarrow{AP} + \overrightarrow{AQ} = $

If $\hat{i}+2 \hat{j}+3 \hat{k}$ and $3 \hat{i}+2 \hat{j}+\hat{k}$ are sides of a parallelogram,then a unit vector parallel to one of the diagonals of the parallelogram is