If $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ are three unit vectors equally inclined to each other at an angle $\alpha$, then the angle between $\overrightarrow{a}$ and the plane of $\overrightarrow{b}$ and $\overrightarrow{c}$ is

If $a$, $b$, $c$ and $A$, $B$, $C\in R-\left\{0\right\}$ such that, $aA+bB+cC+\sqrt{\left(a^2+b^2+c^2\right)\left(A^2+B^2+C^2\right)}=0$, then the value of $\frac{aB}{bA}+\frac{bC}{cB}+\frac{cA}{aC}$ is

If $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},$ $\overrightarrow{a}·\overrightarrow{b}=1$ and $\overrightarrow{a}\times \overrightarrow{b}=\hat{j}-\hat{k},$ then $\overrightarrow{b}·\left(3\hat{i}+4\hat{j}+5\hat{k}\right)=$

if $\overrightarrow{\mathrm{a}},\mathrm{ }\overrightarrow{\mathrm{b}},\mathrm{ }\overrightarrow{\mathrm{c}}$ are unit vectors such that $\overrightarrow{\mathrm{a}}·\overrightarrow{\mathrm{b}}=0,\mathrm{ }\left(\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{c}}\right)·\left(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}\mathrm{ }}\right)=0$ and $\overrightarrow{\mathrm{c}}=\lambda \overrightarrow{\mathrm{a}}+\mu \overrightarrow{\mathrm{b}}+\omega \left(\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}\right)$, where $\lambda , \mu , \omega$ are scalars, then

Let $∆ABC$ be a given triangle. If $\left|\overrightarrow{\mathrm{BA}}-t\overrightarrow{\mathrm{BC}}\right|\ge \left|\overrightarrow{\mathrm{AC}}\right|$ for all $t\in R$, then $∆ABC$ is