Let a be a unit vector perpendicular to the vectors b=i-2j+3k and c=2i+3j-k, and a makes a…

Question

Let $\hat{a}$ be a unit vector perpendicular to the vectors $\vec{b}=\hat{i}-2\hat{j}+3\hat{k}$ and $\vec{c}=2\hat{i}+3\hat{j}-\hat{k}$, and $\hat{a}$ makes an angle of ${\cos }^{-1}(-\frac{1}{3})$ with the vector $\hat{i}+\hat{j}+\hat{k}$. If $\hat{a}$ makes an angle of $\frac{\pi }{3}$ with the vector $\hat{i}+\alpha \hat{j}+\hat{k}$, then the value of $a$ is:

Accepted Answer

$\begin{gathered} \overset{	o }{v}=\hat{i}+\hat{j}+\hat{k} \ \begin{gathered} \overset{	o }{b} \quad 	imes \overset{	o }{c}=|\begin{array}{ccc}\hat{i} \quad \hat{j} \quad \hat{k} \ 1 \quad -2 \quad 3 \ 2 \quad 3 \quad -1\end{array}| \ =-7\hat{i}+7\hat{j}+7\hat{k} \ =-7(\hat{i}-\hat{j}-\hat{k}) \end{gathered}$

Now $\hat{a}=\frac{\hat{i}-\hat{j}-\hat{k}}{\sqrt{3}}$ or $\frac{-\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}$

$\, \, \downarrow \, \, \downarrow$

$ \cos 	heta = \frac{\hat{a}\cdot \overset{	o }{v}}{|\overset{	o }{v}|}=\frac{1-1-1}{\sqrt{3}\sqrt{3}}=\frac{-1}{3}\, \cos 	heta =\frac{\hat{a}\cdot \overset{	o }{v}}{|\overset{	o }{v}|}=\frac{-1+1+1}{3}=\frac{1}{3}\, \ \Rightarrow \hat{a}=\frac{\hat{i}-\hat{j}-\hat{k}}{\sqrt{3}} \end{gathered}$

$\begin{gathered} \cos \frac{\pi }{3}=\frac{\hat{a}\cdot (\hat{i}+\alpha \hat{j}+\hat{k})}{\sqrt{1+{\alpha }^{2}+1}} \ \Rightarrow \frac{1}{2}=\frac{1-\alpha -1}{\sqrt{3}\sqrt{{\alpha }^{2}+2}} \ \Rightarrow \frac{\sqrt{3}}{2}\sqrt{{\alpha }^{2}+2}=-\alpha \, (	herefore \alpha <0) \ 3{\alpha }^{2}+6=4{\alpha }^{2} \ \Rightarrow \alpha =-\sqrt{6} \end{gathered}$

Solutions · 1