Write the distributive law for the product of two vectors. 

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$\vec A$ and $\vec B$ are two vectors and $\theta$ is the angle between them, if $|\vec A \times \vec B|=\sqrt 3(\vec A \cdot \vec B) $ the value of $\theta$ is ......... $^o$

  • [AIPMT 2007]

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 $\theta =$ ........... $^o$

Define the vector product of two vectors.

If $A =a_1 \hat{ i }+b_1 \hat{ j }$ and $B =a_2 \hat{ i }+b_2 \hat{ j }$, the condition that they are perpendicular to each other is

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