(a) If a, b, c and d are four non-zero vectors such that a × b = c × d and a × c =4 b × d, then show that ( a -2 d ) is parallel to (2 b - c ) where…

If $\displaystyle \overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ and $\displaystyle \overrightarrow{\mathrm{d}}$ are four non-zero vectors such that $\displaystyle \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{d}}$ and $\displaystyle \vec{\mathrm{a}} \times \vec{\mathrm{c}}=4 \vec{\mathrm{b}} \times \vec{\mathrm{d}}$, then show that $\displaystyle (\vec{\mathrm{a}}-2 \vec{\mathrm{d}})$ is parallel to $\displaystyle (2 \vec{\mathrm{b}}-\vec{\mathrm{c}})$ where $\displaystyle \vec{\mathrm{a}} \neq 2 \vec{\mathrm{d}}, \vec{\mathrm{c}} \neq 2 \vec{\mathrm{b}}$.

The two adjacent sides of a parallelogram are represented by $\displaystyle 2 \hat{i}-4 \hat{j}-5 \hat{k}$ and $\displaystyle 2 \hat{i}+2 \hat{j}+3 \hat{k}$. Find the unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram also.