If $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$,$|\overrightarrow{a}|=3$,$|\overrightarrow{b}|=5$,and $|\overrightarrow{c}|=7$,then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is

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:

Let $\vec{\alpha}=\hat{i}+\hat{j}+\hat{k}$, $\vec{\beta}=\hat{i}-\hat{j}-\hat{k}$ and $\vec{\gamma}=-\hat{i}+\hat{j}-\hat{k}$ be three vectors. $A$ vector $\vec{\delta}$, in the plane of $\vec{\alpha}$ and $\vec{\beta}$, whose projection on $\vec{\gamma}$ is $\frac{1}{\sqrt{3}}$, is given by

Let $S$ be the set of all $a \in \mathbb{R}$ for which the angle between the vectors $\vec{u} = a(\log_{e} b) \hat{i} - 6 \hat{j} + 3 \hat{k}$ and $\vec{v} = (\log_{e} b) \hat{i} + 2 \hat{j} + 2a(\log_{e} b) \hat{k}$ is acute,where $b > 1$. Then $S$ is equal to:

Let $\bar{a} = \bar{i} + 2\bar{j} + 2\bar{k}$ and $\bar{b} = 2\bar{i} - \bar{j} + p\bar{k}$ be two vectors. If the angle between $\bar{a}$ and $\bar{b}$ is $60^{\circ}$, then $p =$

If the vectors $3i + \lambda j + k$ and $2i - j + 8k$ are perpendicular,then $\lambda$ is: