If the vectors $\bar{a}=\hat{i}-\hat{j}+2 \hat{k}, \bar{b}=2 \hat{i}+4 \hat{j}+\hat{k}$ and…
If the vectors $\bar{a}=\hat{i}-\hat{j}+2 \hat{k}, \bar{b}=2 \hat{i}+4 \hat{j}+\hat{k}$ and $\overline{\mathrm{c}}=\mathrm{p} \hat{\mathrm{i}}+\hat{\mathrm{j}}+\mathrm{q} \hat{\mathrm{k}}$ are mutually orthogonal, then $(\mathrm{p}, \mathrm{q})$ is equal to
$(3,-2)$
$(-2,3)$
$(-3,2)$
$(2,-3)$
Solution
Since the given vectors are mutually orthogonal.
$\therefore \quad \bar{a} \cdot \overline{\mathrm{~b}}=2-4+2=0$
$\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=\mathrm{p}-1+2 \mathrm{q}=0$...(i)
$\overline{\mathrm{b}} \cdot \overline{\mathrm{c}}=2 \mathrm{p}+4+\mathrm{q}=0$...(ii)
Solving (i) and (ii), we get $(p, q)=(-3,2)$