If the line $\bar{r}=(\hat{\imath}-2 \hat{\jmath}+3 \hat{k})+\lambda(2 \hat{\imath}+\hat{\jmath}+2 \hat{k})$…
If the line $\bar{r}=(\hat{\imath}-2 \hat{\jmath}+3 \hat{k})+\lambda(2 \hat{\imath}+\hat{\jmath}+2 \hat{k})$ is parallel to the plane $\bar{r} .(3 \hat{\imath}-2 \hat{\jmath}+m \hat{k})=10$, then the value of $\mathrm{m}$ is
2
-3
-2
3
Solution
$\overline{\mathrm{b}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ and $\overline{\mathrm{n}}=3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\mathrm{m} \hat{\mathrm{k}}$
Since, line is parallel to the plane, $\bar{b} \cdot \bar{n}=0$
$(\hat{2} \mathrm{i}+\hat{\mathrm{j}}+2 \hat{\mathrm{k}}) \cdot(3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\mathrm{m} \hat{\mathrm{k}}) \quad=0 \Rightarrow 6-2+2 \mathrm{~m}=0 \Rightarrow \mathrm{m}=-2$