If the position vectors of the vertices $A, B$ and $C$ of $\triangle ABC$ are $\hat{i}+2\hat{j}-5\hat{k}$,$-2\hat{i}+2\hat{j}+\hat{k}$ and $2\hat{i}+\hat{j}-\hat{k}$ respectively,then $\angle B=$

The value of $\lambda$ for which the vectors $\vec{a} = 2\hat{i} + \lambda\hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} + 3\hat{k}$ are orthogonal is

Let $ABC$ be a triangle. Let $u = \overrightarrow{AB}$ and $v = \overrightarrow{AC}$. If $D$ is the midpoint of $BC$,then the length of the median of $\triangle ABD$ through the vertex $B$ is:

$A$ vector whose modulus is $\sqrt{51}$ and makes the same angle with $a = \frac{i - 2j + 2k}{3}$,$b = \frac{-4i - 3k}{5}$,and $c = j$ is:

If $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$, $|\vec{a}|=3$, $|\vec{b}|=5$, and $|\vec{c}|=7$, then the angle between $\vec{a}$ and $\vec{b}$ is

Let the position vectors of the vertices of a triangle $ABC$ be $\bar{a}, \bar{b}, \bar{c}$. If on the plane of the triangle,$P$ is a point having position vector $\bar{x}$ such that $\bar{x} \cdot (\bar{c} - \bar{b}) = \bar{a} \cdot \bar{c} - \bar{a} \cdot \bar{b}$ and $\bar{x} \cdot (\bar{a} - \bar{c}) = \bar{a} \cdot \bar{b} - \bar{b} \cdot \bar{c}$,then for the triangle $ABC$,$P$ is the