Question

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

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

Answer 1

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

The vector along the bisector of the given angle is
$\overrightarrow{c}=t(\frac{7\hat{i}-4\hat{j}-4\hat{k}}{\sqrt{81}}+\frac{-2\hat{i}-\hat{j}+2\hat{k}}{\sqrt{9}})$
$\overrightarrow{c}=t\left(\frac{i-7j+2k}{9}\right)$
Since, $|\overrightarrow{c}|=5\sqrt{6},5\sqrt{6}=t\frac{\sqrt{54}}{9}$
$\Rightarrow t=15$
$\Rightarrow \overrightarrow{c}=\frac{5}{3}(\hat{i}-7\hat{j}+2\hat{k})$