• # question_answer The vector c directed along the internal bisector of the angle between the vectors $\mathbf{a}=7\mathbf{i}-4\mathbf{j}-4\mathbf{k}$ and $\mathbf{b}=-2\mathbf{i}-\mathbf{j}+2\mathbf{k}$ with $|\mathbf{c}|\,=5\sqrt{6},$ is    A) $\frac{5}{3}\,(\mathbf{i}-7\mathbf{j}+2\mathbf{k})$                     B) $\frac{5}{3}\,(5\mathbf{i}+5\mathbf{j}+2\mathbf{k})$ C) $\frac{5}{3}\,(\mathbf{i}+7\mathbf{j}+2\mathbf{k})$       D) $\frac{5}{3}\,(-5\mathbf{i}+5\mathbf{j}+2\mathbf{k})$

Solution :

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

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