If $a = i + 2j - 3k$ and $b = 3i - j + 2k,$ then the angle between the vectors $a + b$ and $a - b$ is ............... $^o$

If $\overline{a}, \overline{b}, \overline{c}$ are three vectors such that $\overline{a} \cdot(\overline{b}+\overline{c})+\overline{b} \cdot(\overline{c}+\overline{a})+\overline{c} \cdot(\overline{a}+\overline{b})=0$ and $|\overline{a}|=1$,$|\overline{b}|=8$ and $|\overline{c}|=4$,then $|\overline{a}+\overline{b}+\overline{c}|$ has the value

If the vertices $A, B, C$ of a triangle $ABC$ are $(1,2,3), (-1,0,0), (0,1,2)$ respectively,then find $\angle ABC$. $[\angle ABC \text{ is the angle between the vectors } \overrightarrow{BA} \text{ and } \overrightarrow{BC}]$.

The position vectors of the vertices of a $\Delta ABC$ are $4\hat{i}-2\hat{j}$,$\hat{i}+4\hat{j}-3\hat{k}$,and $-\hat{i}+5\hat{j}+\hat{k}$ respectively. Then,$\angle ABC$ is equal to:

If the vectors $\vec{a} = \hat{i} - \hat{j} + 2\hat{k}$,$\vec{b} = 2\hat{i} + 4\hat{j} + \hat{k}$,and $\vec{c} = \alpha\hat{i} + \hat{j} + \beta\hat{k}$ are mutually orthogonal,then $(\alpha, \beta) = $

Let $\overrightarrow{a}=2 \hat{i}-\hat{j}+\hat{k}$,$\overrightarrow{b}=\hat{i}+2 \hat{j}-\hat{k}$ and $\overrightarrow{c}=\hat{i}+\hat{j}-2 \hat{k}$ be three vectors. $A$ vector in the plane of $\overrightarrow{b}$ and $\overrightarrow{c}$ whose projection on $\overrightarrow{a}$ is of magnitude $\sqrt{\frac{2}{3}}$,is