If $\vec A$ and $\vec B$ are perpendicular vectors and vector $\vec A = 5\hat i + 7\hat j - 3\hat k$ and $\vec B = 2\hat i + 2\hat j - a\hat k.$ The value of $a$ is

Define the scalar product and obtain the magnitude of a vector from it. Mention the direction of scalar product.

The position vectors of points $A, B, C$ and $D$ are $A = 3\hat i + 4\hat j + 5\hat k,\,\,B = 4\hat i + 5\hat j + 6\hat k,\,\,C = 7\hat i + 9\hat j + 3\hat k$ and $D = 4\hat i + 6\hat j$ then the displacement vectors $AB$ and $CD $ are

$\overrightarrow A = 2\hat i + 4\hat j + 4\hat k$ and $\overrightarrow B = 4\hat i + 2\hat j - 4\hat k$ are two vectors. The angle between them will be ........ $^o$

If for two vector $\overrightarrow A $ and $\overrightarrow B $, sum $(\overrightarrow A + \overrightarrow B )$ is perpendicular to the difference $(\overrightarrow A - \overrightarrow B )$. The ratio of their magnitude is

Two forces ${\vec F_1} = 5\hat i + 10\hat j - 20\hat k$ and ${\vec F_2} = 10\hat i - 5\hat j - 15\hat k$ act on a single point. The angle between ${\vec F_1}$ and ${\vec F_2}$ is nearly ....... $^o$