Question

The vector $\overline{c}$ directed along the internal bisector of the angle between the vectors $\overline{a}=7\hat{i}-4\hat{j}-4\hat{k}$ and $\overline{b}=-2\hat{i}-\hat{j}+2\hat{k}$ with $\left|\overline{c}\right|=3\sqrt{6}$

• A. $\frac{(3\hat{i}-7\hat{j})+2\hat{k}}{9}$
• B. $\frac{5(5\hat{i}+5\hat{j}+2\hat{k})}{3}$
• C. $5\frac{(\hat{i}-7\hat{j}+2\hat{k})}{3}$
• D. $5(-5\hat{j}+5\hat{j}+2\hat{k})$

Answer 1

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

The internal angle bisector is given by adding the unit vectors along both the arms of the angle.
Thus we get $\frac{7\hat{i}-4\hat{j}-4\hat{k}}{9}+\frac{-2\hat{i}-\hat{j}+2\hat{k}}{3}=\frac{\hat{i}-7\hat{j}+2\hat{k}}{9}$
Now for the magnitude to be $5\sqrt{6}$, we have $5\sqrt{6}\times \frac{\hat{i}-7\hat{j}+2\hat{k}}{3\sqrt{6}}=\frac{5}{3}(\hat{i}-7\hat{j}+2\hat{k})$