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

Let $\left| {{{\vec A}_1}} \right| = 3,\,\left| {\vec A_2} \right| = 5$, and $\left| {{{\vec A}_1} + {{\vec A}_2}} \right| = 5$. The value of $\left( {2{{\vec A}_1} + 3{{\vec A}_2}} \right)\cdot \left( {3{{\vec A}_1} - 2{{\vec A}_2}} \right)$ is

The components of $\vec a = 2\hat i + 3\hat j$ along the direction of vector $\left( {\hat i + \hat j} \right)$ is

Show that the area of the triangle contained between the vectors $a$ and $b$ is one half of the magnitude of $a \times b .$

If diagonals of a parallelogram are $\left( {5\hat i - 4\hat j + 3\hat k} \right)$ and $\left( {3\hat i + 2\hat j - \hat k} \right)$ then its area is