Question

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

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

Answer 1

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

The required vectors $c$ is given by

$c=\lambda (a+b)=\lambda (\frac{a}{\left|a\right|}+\frac{b}{\left|b\right|})=\lambda \{\frac{1}{9}(7i-4j-4k)+\frac{1}{3}(-2i-j+2k)\}$

$\Rightarrow c=\frac{\lambda }{9}(i-7j+2k)$

$\Rightarrow \left|c\right|=\pm \frac{\lambda }{9}\sqrt{1+49+4}=\pm \frac{\lambda }{9}\sqrt{54}$

But $\left|c\right|=5\sqrt{6}$

$\therefore \pm \frac{\lambda }{9}\sqrt{54}=5\sqrt{6}\Rightarrow \lambda =\pm 15$

Hence, $c=\pm \frac{15}{9}(i-7j+2k)=\pm \frac{5}{3}(i-7j+2k)$