If $\vec{a} = \hat{i}+\hat{j}-\hat{k}$ and $\vec{b} = 2\hat{i}-3\hat{j}+\hat{k}$ are adjacent sides of a parallelogram,then the lengths of its diagonals are

If the points with position vectors $10\hat{i} + 3\hat{j}$,$12\hat{i} - 5\hat{j}$,and $a\hat{i} + 11\hat{j}$ are collinear,then the value of $a$ is:

The direction cosines of the vector $\hat{i}-2\hat{j}+3\hat{k}$ are . . . . . . .

Find the position vector of a point $R$ which divides the line joining two points $P$ and $Q$ whose position vectors are $(2 \vec{a}+\vec{b})$ and $(\vec{a}-3 \vec{b})$ externally in the ratio $1:2$. Also,show that $P$ is the mid-point of the line segment $RQ$.

If three points $A$,$B$,and $C$ have position vectors $(1, x, 3)$,$(3, 4, 7)$,and $(y, -2, -5)$ respectively and if they are collinear,then $(x, y)$ is

If $P$ divides the line segment joining the points $A$ and $B$ in the ratio $2:1$ and the position vectors of $A$ and $B$ are $\hat{i}-2\hat{j}$ and $-3\hat{i}+5\hat{j}$ respectively,then the position vector of $P$ is