If a vector $2\hat i + 3\hat j + 8\hat k$ is perpendicular to the vector $4\hat j - 4\hat i + \alpha \hat k$. Then the value of $\alpha $ is

If $|\vec A \times \vec B| = \sqrt 3 \vec A.\vec B,$ then the value of$|\vec A + \vec B|$ is

Which of the following is the unit vector perpendicular to $\overrightarrow A $ and $\overrightarrow B $

Force $F$ applied on a body is written as $F =(\hat{ n } \cdot F ) \hat{ n }+ G$, where $\hat{ n }$ is a unit vector. The vector $G$ is equal to

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 two vectors have magnitudes $3$ and $5$. If angle between them is $60^o$, then the dot product of two vectors will be