Question

Find a unit vector perpendicular to both the vectors
$(2 \hat{ i }+3 \hat{ j }+\hat{ k })$ and $(\hat{ i }-\hat{ j }+2 \hat{ k })$

• A. $\frac{1}{\sqrt{8}}(4 \hat{i}+3 \hat{j}-5 \hat{k})$
• B. $\frac{1}{\sqrt{83}}(7 \hat{i}-3 \hat{j}-5 \hat{k})$
• C. $\frac{1}{\sqrt{8}}(4 \hat{i}+3 \hat{j}+1 \hat{k})$
• D. None of these

Answer 1

Answer: (B)

Let $\overrightarrow{ A }=2 \hat{ i }+3 \hat{ j }+\hat{ k }$ and
$\overrightarrow{ B }=\hat{ i }-\hat{ j }+2 \hat{ k }$ unit vector
perpendicular to both $\overrightarrow{ A }$ and
$\overrightarrow{ B }$ is $n =\frac{\overrightarrow{ A } \times \overrightarrow{ B }}{|\overrightarrow{ A } \times \overrightarrow{ B }|}$
$\overrightarrow{ A } \times \overrightarrow{ B }=\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\ 2&3&1\\ 1&-1&2\end{vmatrix}$
$=\hat{ i }(6+1)-\hat{ j }(4-1)+\hat{ k }(-2-3)$
$=7 \hat{ i }-3 \hat{ j }-5 \hat{ k }$
$\therefore |\overrightarrow{ A } \times \overrightarrow{ B }|=\sqrt{7^{2}+(-3)^{2}+(-5)^{2}}$
$=\sqrt{83} unit$
$\therefore \hat{ n }=\frac{1}{\sqrt{83}}$
$(7 \hat{ i }-3 \hat{ j }-5 \hat{ k })$