(a) Three honey bees were found flying along the vectors a =2 i -3 j + k, b =4 j -2 k and c =3 i +2 k respectively. Find the value of λ such that the…

Three honey bees were found flying along the vectors \[\vec{\mathrm{a}}=2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+\hat{\mathrm{k}}, \vec{\mathrm{b}}=4 \hat{\mathrm{j}}-2 \hat{\mathrm{k}} \text { and } \vec{\mathrm{c}}=3 \hat{\mathrm{i}}+2 \hat{\mathrm{k}} \text { respectively. } \] Find the value of $\displaystyle \lambda$ such that the path for $\displaystyle \overrightarrow{\mathrm{a}}+\lambda \overrightarrow{\mathrm{b}}$ is perpendicular to $\displaystyle \overrightarrow{\mathrm{c}}$.

If A, B and C be three non-collinear points such that $\displaystyle \overrightarrow{\mathrm{AB}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\displaystyle \overrightarrow{\mathrm{AC}}=2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}$, then find the area of $\displaystyle \triangle \mathrm{ABC}$.