$A$ unit vector perpendicular to the plane containing the vectors $\hat{i}+2\hat{j}+\hat{k}$ and $-2\hat{i}+\hat{j}+3\hat{k}$ is

Let $\overrightarrow{a}=\hat{i}+2 \hat{j}+3 \hat{k}$ and $\overrightarrow{b}=\hat{i}-2 \hat{j}-3 \hat{k}$ be two vectors. If $A_1$ is the area of the quadrilateral having $\vec{a}, \vec{b}$ as its diagonals and $A_2$ is the area of the parallelogram having $\overrightarrow{a}, \overrightarrow{b}$ as its two adjacent sides,then $A_1 \cdot A_2=$

Prove that in a $\Delta ABC$,$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$,where $a, b, c$ represent the magnitudes of the sides opposite to vertices $A, B, C$,respectively.

Let vectors $a, b, c$ and $d$ be such that $(a \times b) \times (c \times d) = 0$. If $a$ and $b$ lie in plane $P_1$ and $c$ and $d$ lie in plane $P_2$,find the angle between $P_1$ and $P_2$.

If $A, B, C, D$ are four points in space,then $|\overline{AB} \times \overline{CD} + \overline{BC} \times \overline{AD} + \overline{CA} \times \overline{BD}| = \lambda \times (\text{Area of } \Delta ABC)$. Find $\lambda$.

If $a$ and $b$ are unit vectors,then the vector $(a+b) \times (a \times b)$ is parallel to the vector