Consider three vectors $A =\hat{ i }+\hat{ j }-2 \hat{ k }, B =\hat{ i }-\hat{ j }+\hat{ k }$ and $C =2 \hat{ i }-3 \hat{ j }+4 \hat{ k }$. A vector $X$ of the form $\alpha A +\beta B$ ( $\alpha$ and $\beta$ are numbers) is perpendicular to $C$.The ratio of $\alpha$ and $\beta$ is

A particle moves in the $x-y$ plane under the action of a force $\overrightarrow F $ such that the value of its linear momentum $(\overrightarrow P )$ at anytime t is ${P_x} = 2\cos t,\,{p_y} = 2\sin t.$ The angle $\theta $between $\overrightarrow F $ and $\overrightarrow P $ at a given time $t$. will be $\theta =$ ........... $^o$

The angle between two vectors $4\hat i + 3\hat j + \hat k$ and $-3\hat i + 2\hat j + 6\hat k$ is ....... $^o$

If $\vec A = 2\hat i + \hat j - \hat k,\,\vec B = \hat i + 2\hat j + 3\hat k$ and $\vec C = 6\hat i - 2j - 6\hat k$ then the angle between $(\vec A + \vec B)$ and $\vec C$ wil be ....... $^o$

For three vectors $\vec{A}=(-x \hat{i}-6 \hat{j}-2 \hat{k})$, $\vec{B}=(-\hat{i}+4 \hat{j}+3 \hat{k})$ and $\vec{C}=(-8 \hat{i}-\hat{j}+3 \hat{k})$, if $\overrightarrow{\mathrm{A}} \cdot(\overrightarrow{\mathrm{B}} \times \overrightarrow{\mathrm{C}})=0$, them value of $\mathrm{x}$ is. . . . . .. 

The value of $(\overrightarrow A + \overrightarrow B )\, \times (\overrightarrow A - \overrightarrow B )$ is