If three vectors $a, b, c$ satisfy $a + b + c = 0$ and $|a| = 3, |b| = 5, |c| = 7,$ then the angle between $a$ and $b$ is .............. $^o$

If $4 \hat{i}+7 \hat{j}+8 \hat{k}$,$2 \hat{i}+3 \hat{j}+4 \hat{k}$ and $2 \hat{i}+5 \hat{j}+7 \hat{k}$ are the position vectors of the vertices $A$,$B$ and $C$ respectively of triangle $ABC$,then the position vector of the point in which the bisector of $\angle B$ meets $CA$ is:

If $a$ and $b$ are respectively the internal and external bisectors of the angles between the vectors $u = -\hat{i} + 2\hat{j} - 2\hat{k}$ and $v = 3\hat{i} + 4\hat{j}$,and $|a| = \frac{2}{3}\sqrt{6}$,$|b| = \frac{2}{3}\sqrt{3}$,then one of the values of $a - b$ is

Let $\sqrt{3} \hat{i} + \hat{j}$,$\hat{i} + \sqrt{3} \hat{j}$ and $\beta \hat{i} + (1 + \beta) \hat{j}$ respectively be the position vectors of the points $A, B$ and $C$ with respect to the origin $O$. If the distance of $C$ from the bisector of the acute angle between $OA$ and $OB$ is $\frac{3}{\sqrt{2}}$,then the sum of all possible values of $\beta$ is:

If $\bar{a}$ and $\bar{b}$ are two unit vectors such that $\bar{a}+2 \bar{b}$ and $5 \bar{a}-4 \bar{b}$ are perpendicular to each other,then the angle between $\bar{a}$ and $\bar{b}$ is

If the angle between the vectors $2 \alpha^2 \hat{i} + 4 \alpha \hat{j} + \hat{k}$ and $7 \hat{i} - 2 \hat{j} + \alpha \hat{k}$ is obtuse,then