Let $ABC$ be a triangle. Let $u = \vec{AB}$ and $v = \vec{AC}$. If $D$ is the midpoint of $BC$,then $\vec{AD} =$

If in a triangle $\overrightarrow{AB} = \vec{a}$ and $\overrightarrow{AC} = \vec{b}$,and $D$ and $E$ are the mid-points of $AB$ and $AC$ respectively,then $\overrightarrow{DE}$ is equal to:

Let $\overrightarrow{OA} = \hat{i} - 3\hat{j} + \hat{k}$,$\overrightarrow{OB} = \hat{i} + 3\hat{j} - 2\hat{k}$,and $\overrightarrow{OC} = 4\hat{i} + 3\hat{j} + 5\hat{k}$ be the position vectors of three points $A$,$B$,and $C$. Let $P$ be the point which divides $AB$ in the ratio $2:1$. If $l, m, n$ are the direction cosines of the vector $\overrightarrow{PC}$,then $l + 3m + 2n =$

Classify the following measure as a scalar or a vector:
$40^{o}$

If the length of a vector is $21$ and its direction ratios are $2, -3, 6$,then its direction cosines are:

If $a = 3i - 2j + k$,$b = 2i - 4j - 3k$ and $c = -i + 2j + 2k$,then $a + b + c$ is