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

If the vectors$\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ are $a_1\hat{i} + a_2\hat{j} +a_3 \hat{k} , b_1\hat{i} +b_2\hat{j} +b_3\hat{k}$ and $c_1\hat{i}+c_2\hat{j}+c_3\hat{k}$ respectively, then show that $\overrightarrow{a}\times (\overrightarrow{b}+\overrightarrow{c})=\overrightarrow{a}\times \overrightarrow{b} +\overrightarrow{a}\times \overrightarrow{c}.$

If$\left|\overrightarrow{a}\right|=13, \left|\overrightarrow{b}\right|=5$ and $\overrightarrow{a}·\overrightarrow{b}= 60$, then find$\left|\overrightarrow{a}\times \overrightarrow{b}\right|$.

If the volume of the parallelepiped whose coterminous edges are represented by the vectors $5\hat{i}-4\hat{j}+\hat{k}, 4\hat{i}+3\hat{j}+\lambda \hat{k}$ and $\hat{i}-2\hat{j}+7\hat{k}$ is $216$ cubic units, find the value of $\lambda$.

Using scalar triple product, prove that the points $\left(-1, 4, 3\right), \left(3, 2, 5\right), \left(-3, 8, -5\right)$ and $\left(-3, 2, 1\right)$ are coplanar.

The position vectors of the points $A, B, C$ and $D$ are $3\hat{i}-2\hat{j}-\hat{k}, 2\hat{i}+3\hat{j}-4\hat{k}, -\hat{i}+\hat{j}+2\hat{k}$ and $4\hat{i}+5\hat{j}+\lambda \hat{k}$ respectively. If the points $A, B, C$ and $D$ lie in a plane, then find the value of $\lambda$.