Question

The vector c directed along the internal bisector of the angle between the vectors $a=7\hat{i}-4\hat{j}-4\hat{k}$ and $b=-2\hat{i}-\hat{j}+2\hat{k}$ with $|c|=5\sqrt{6}$, is

• A. $\frac{5}{3}(\hat{i}-7\hat{j}+2\hat{k})$
• B. $\frac{5}{3}(\hat{i}+7\hat{j}+2\hat{k})$
• C. $\frac{5}{3}(5\hat{i}+5\hat{j}+2\hat{k})$

Answer 1

The correct option is A $\frac{5}{3}(\hat{i}-7\hat{j}+2\hat{k})$

The required vector c is given by $\lambda (\frac{a}{|a|}+\frac{b}{|b|})$
Now, $\frac{a}{|a|}=\frac{1}{9}(7\hat{i}-4\hat{j}-4\hat{k})$
and $\frac{b}{|b|}=\frac{1}{3}(-2\hat{i}-\hat{j}+2\hat{k})\Rightarrow c=\lambda (\frac{1}{9}\hat{i}-\frac{7}{9}\hat{j}+\frac{2}{9}\hat{k})\Rightarrow |c{|}^2={\lambda }^2.\frac{54}{81}$
$\Rightarrow (5\sqrt{6}{)}^2={\lambda }^2.\frac{54}{81}$
$\Rightarrow 150={\lambda }^2.\frac{54}{81}$
$\Rightarrow {\lambda }^2=225or\lambda \pm 15.$
Therefore, $c=\pm \frac{5}{3}(\hat{i}-7\hat{j}+2\hat{k}).$