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 $\vec{\alpha}=4 \hat{i}+3 \hat{j}+5 \hat{k}$ and $\vec{\beta}=\hat{i}+2 \hat{j}-4 \hat{k}$. Let $\vec{\beta}_1$ be parallel to $\vec{\alpha}$ and $\vec{\beta}_2$ be perpendicular to $\vec{\alpha}$. If $\vec{\beta}=\vec{\beta}_1+\vec{\beta}_2$,then the value of $5 \vec{\beta}_2 \cdot(\hat{i}+\hat{j}+\hat{k})$ is

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

If $\overline{e_1}, \overline{e_2}$ are two non-collinear unit vectors such that $|\overline{e_1}+\overline{e_2}|=\sqrt{3}$,then $(2 \overline{e_1}-5 \overline{e_2}) \cdot (3 \overline{e_1}+\overline{e_2}) = $

If $a+b+c=0$ and $|a|=3, |b|=5, |c|=7$,then the angle between $a$ and $b$ is ........ (in $^{\circ}$)

Let $\overrightarrow{x}$ be a vector in the plane containing vectors $\overrightarrow{a} = 2\hat{i} - \hat{j} + \hat{k}$ and $\overrightarrow{b} = \hat{i} + 2\hat{j} - \hat{k}$. If the vector $\overrightarrow{x}$ is perpendicular to $(3\hat{i} + 2\hat{j} - \hat{k})$ and its projection on $\overrightarrow{a}$ is $\frac{17\sqrt{6}}{2}$,then the value of $|\overrightarrow{x}|^{2}$ is equal to ...... .