The two adjacent sides of a parallelogram are represented by vectors $2\hat{\mathit{i}}-4\hat{\mathit{j}}+$ $5\hat{\mathit{k}}$ and $\hat{\mathit{i}}-2\hat{\mathit{j}}-3\hat{\mathit{k}}$. Find the unit vector parallel to one of its diagonals. Also, find the area of the parallelogram.
OR
If $\overrightarrow{\mathit{a}}=2\hat{\mathit{i}}+2\hat{\mathit{j}}+3\hat{\mathit{k}},\overrightarrow{\mathit{b}}=-\hat{\mathit{i}}+2\hat{\mathit{j}}+\hat{\mathit{k}}$ and $\overrightarrow{\mathit{c}}=3\hat{\mathit{i}}+\hat{\mathit{j}}$ are such that the vector $\left(\overrightarrow{\mathit{a}}+\mathit{\lambda }\overrightarrow{\mathit{b}}\right)$ is perpendicular to vector $\overrightarrow{\mathit{c}}$, then find the value of $\mathit{\lambda }$.