If in a right-angled triangle $ABC$,the hypotenuse $|\overrightarrow{AB}| = p$,then $\overrightarrow{AB} \cdot \overrightarrow{AC} + \overrightarrow{BC} \cdot \overrightarrow{BA} + \overrightarrow{CA} \cdot \overrightarrow{CB} = $

If $\vec{a}$ is perpendicular to $\vec{b}$ and $\vec{r}$ is a non-zero vector such that $p\vec{r} + (\vec{r} \cdot \vec{b})\vec{a} = \vec{c}$,then $\vec{r} = $

If $\overline{p}=2 \hat{i}+\hat{k}$,$\overline{q}=\hat{i}+\hat{j}+\hat{k}$,$\overline{r}=4 \hat{i}-3 \hat{j}+7 \hat{k}$ and a vector $\overline{m}$ is such that $\overline{m} \times \overline{q}=\overline{r} \times \overline{q}$ and $\overline{m} \cdot \overline{p}=0$,then $\overline{m} = \dots$

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

If the angle between $\bar{a} = x\bar{i} - 3\bar{j} - \bar{k}$ and $\bar{b} = 2x\bar{i} + x\bar{j} - \bar{k}$ is acute and the angle between $\bar{b}$ and the $y$-axis is obtuse,then $x$ belongs to which interval?

If $a=\hat{i}+\hat{j}+\hat{k}$,$a \cdot b=1$ and $a \times b=\hat{j}-\hat{k}$,then $b=$