Let $P(3, 2, 3)$,$Q(4, 6, 2)$,and $R(7, 3, 2)$ be the vertices of $\triangle PQR$. Then,the angle $\angle QPR$ is

If $P=(0,1,2), Q=(4,-2,1)$ and $O=(0,0,0)$,then $\angle POQ=$

$A(\vec{a}), B(\vec{b}), C(\vec{c}), D(\vec{d})$ are four concyclic points, such that $x \vec{a}+y \vec{b}+z \vec{c}+t \vec{d}=\vec{0}$ and $x+y+z+t=0$, where $x, y, z, t$ are constants not all zero. If the chords $AB$ and $CD$ intersect at $P$, then:

The angle between two vectors $\vec{a} = \hat{i} - \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + \hat{j} - \hat{k}$ is . . . . . . .

Let the vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ be such that $|\overrightarrow{a}|=2, |\overrightarrow{b}|=4$ and $|\overrightarrow{c}|=4$. If the projection of $\overrightarrow{b}$ on $\overrightarrow{a}$ is equal to the projection of $\overrightarrow{c}$ on $\overrightarrow{a}$ and $\overrightarrow{b}$ is perpendicular to $\overrightarrow{c}$,then the value of $|\overrightarrow{a}+\overrightarrow{b}-\overrightarrow{c}|$ is

If $\vec{a}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}+3\hat{k}$,then $(\vec{a}+\vec{b}) \cdot (\vec{a}-\vec{b}) = $ . . . . . . .