The angle between the two vectors $\vec A = 3\hat i + 4\hat j + 5\hat k$ and $\vec B = 3\hat i + 4\hat j - 5\hat k$ will be....... $^o$

If $\overrightarrow {\rm A} = 2\hat i + 3\hat j - \hat k$ and $\overrightarrow B = - \hat i + 3\hat j + 4\hat k$ then projection of $\overrightarrow A $ on $\overrightarrow B $ will be

Obtain scalar product in terms of Cartesian component of vectors.

Let $\vec A\, = \,(\hat i\, + \,\hat j)\,$  and $\vec B\, = \,(2\hat i\, - \,\hat j)\,.$  The magnitude of a coplanar vector $\vec C$ such that  $\vec A\cdot \vec C\, = \,\vec B\cdot \vec C\, = \vec A\cdot \vec B$ is given by

A vector has magnitude same as that of $\overrightarrow{\mathrm{A}}-=3 \hat{\mathrm{j}}+4 \hat{\mathrm{j}}$ and is parallel to $\overrightarrow{\mathrm{B}}=4 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}$. The $\mathrm{x}$ and $y$ components of this vector in first quadrant are $\mathrm{x}$ and $3$ respectively where  $X$=_____.

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