The angle between vectors $(\vec{M} \times \vec{N})$ and $(\vec{N} \times \vec{M})$ is ................ (in $^{\circ}$)

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

Given $|\vec{A}_1| = 2$,$|\vec{A}_2| = 3$,and $|\vec{A}_1 + \vec{A}_2| = 3$. Find the value of $|(\vec{A}_1 + 2\vec{A}_2) \times (3\vec{A}_1 - 4\vec{A}_2)|$.

If $\vec A = 2\hat i + \hat j - \hat k$,$\vec B = \hat i + 2\hat j + 3\hat k$ and $\vec C = 6\hat i - 2\hat j - 6\hat k$,then the angle between $(\vec A + \vec B)$ and $\vec C$ will be ....... $^o$.

The angle between the vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ is $\theta$. The value of the triple product $\overrightarrow{A} \cdot (\overrightarrow{B} \times \overrightarrow{A})$ is

If $|\overrightarrow{A}|=4$ unit,$|\overrightarrow{A}+\overrightarrow{B}|=10$ unit and $\overrightarrow{A} \cdot(\overrightarrow{A}+\overrightarrow{B})=20$ unit,then $|\overrightarrow{B}|=$ ?