Define the vector product of two vectors.

If $A$ is a unit vector in a given direction, then the value of $\hat{ A } \cdot \frac{d \hat{ A }}{d t}$ is

The area of the parallelogram determined by $A =2 \hat{ i }+\hat{ j }-3 \hat{ k }$ and $B =12 \hat{ j }-2 \hat{ k }$ is approximately

Given : $\vec A\, = \,2\hat i\, + \,p\hat j\, + q\hat k$ and $\vec B\, = \,5\hat i\, + \,7\hat j\, + 3\hat k,$ if $\vec A\,||\,\vec B,$ then the values of $p$ and $q$ are, respectively

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