If the vectors $\overline{\mathrm{a}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}},…
If the vectors $\overline{\mathrm{a}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}, \overline{\mathrm{b}}=2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overline{\mathrm{c}}=\lambda \hat{\mathrm{i}}+\hat{\mathrm{j}}+\mu \hat{\mathrm{k}}$ are mutually orthogonal, then $(\lambda, \mu) \equiv$
$(-3,2)$
$(-2,3)$
$(2,-3)$
$(3,-2)$
Solution
As vectors $\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ are mutually orthogonal, we get
$\begin{array}{ll}
& \overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=0 \\
\therefore & \lambda-1+2 \mu=0 \\
\therefore \quad & \lambda+2 \mu=1...(i)
\end{array}$
and $\bar{b} \cdot \bar{c}=0$
$\begin{array}{ll}
\therefore & 2 \lambda+4+\mu=0 \\
\therefore & 2 \lambda+\mu=-4...(ii)
\end{array}$ Solving (i) and (ii), we get
$\lambda=-3, \mu=2$