Let $D, E, F$ be the midpoints of the sides $BC, CA, AB$ of a triangle $ABC$ respectively. Then $\vec{AD} + \vec{BE} + \vec{CF} = \dots$

In $\triangle PQR$,$(4 \hat{i}+3 \hat{j}+6 \hat{k})$,$(2 \hat{i}+2 \hat{j}+3 \hat{k})$ and $(3 \hat{i}+\hat{j}+3 \hat{k})$ are the position vectors of the vertices $P, Q$ and $R$ respectively. Then the position vector of the point of intersection of the angle bisector of $P$ with $QR$ is

If a vector $\vec{v} = 3\hat{i} + 4\hat{j} - 5\hat{k}$ is rotated about the origin, its magnitude remains constant. If the new vector is $\vec{v}' = (a + 1)\hat{i} - 3\hat{j} + 5\hat{k}$, find the possible values of $a$.

The ratio in which $\hat{i}+2 \hat{j}+3 \hat{k}$ divides the join of $-2 \hat{i}+3 \hat{j}+5 \hat{k}$ and $7 \hat{i}-\hat{k}$ is

If a vector $\vec{r}$ makes equal angles with the axes $OX, OY,$ and $OZ$,find the total number of such vectors $\vec{r}$.

If the angle between $\vec{a}$ and $\vec{b}$ is $30^o$,then the angle between $3\vec{a}$ and $-4\vec{b}$ will be ............ $^o$.