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

If a vector $\vec A$ is parallel to another vector $\vec B$,then the resultant of the vector $\vec A \times \vec B$ will be equal to

If $\vec{A} \times \vec{B} = \vec{B} \times \vec{C} = \vec{C} \times \vec{A}$,then $\vec{A} + \vec{B} + \vec{C}$ is equal to:

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

If $\overrightarrow{A}$ and $\overrightarrow{B}$ are two vectors,then which of the following are correct?
$(a) \ (\overrightarrow{A} \times \overrightarrow{B}) \perp \overrightarrow{A}$
$(b) \ (\overrightarrow{A} \times \overrightarrow{B}) \perp \overrightarrow{B}$
$(c) \ (\overrightarrow{A} \times \overrightarrow{B}) \perp (\overrightarrow{A} + \overrightarrow{B})$
$(d) \ (\overrightarrow{A} \times \overrightarrow{B}) \perp (\overrightarrow{A} - \overrightarrow{B})$
$(e) \ (\overrightarrow{A} \times \overrightarrow{B}) \perp (\overrightarrow{A} \cdot \overrightarrow{B})$

Find the scalar and vector products of two vectors $\vec{a} = (3 \hat{i} - 4 \hat{j} + 5 \hat{k})$ and $\vec{b} = (-2 \hat{i} + \hat{j} - 3 \hat{k})$.