Question

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

Answer 1

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

The required vector $c$ is given by $\lambda (\frac{a}{\left|a\right|}+\frac{b}{\left|b\right|})$
Now $\frac{a}{\left|a\right|}=\frac{1}{9}(7i-4j-4k)$
and $\frac{b}{\left|b\right|}=\frac{1}{3}(-2i-j+2k)$
$\Rightarrow c=\lambda (\frac{1}{9}i-\frac{7}{9}j+\frac{2}{9}k)\Rightarrow {\left|c\right|}^2={\lambda }^2\times \frac{27}{81}\times 2$
Therefore ${\lambda }^2=225$ or $\lambda =\pm 15$
$\Rightarrow c=\pm \frac{5}{3}(i-7j+2k)$