• # question_answer The vector $\mathbf{a}+\mathbf{b}$ bisects the angle between the vectors a and b, if A) $|\mathbf{a}|\,=\,|\mathbf{b}|$ B) $|\mathbf{a}|\,=\,|\mathbf{b}|$ or angle between a and b is zero C) $|\mathbf{a}|\,\,=m\,|\mathbf{b}|$ D) None of these

• Since the angle between $\mathbf{a}+\mathbf{b}$ and $\mathbf{a}$and the angle between $\mathbf{a}+\mathbf{b}$ and $\mathbf{b}$are the same, so we have
• $\frac{(\mathbf{a}+\mathbf{b})\,.\,\mathbf{a}}{|\mathbf{a}+\mathbf{b}|\,|\mathbf{a}|}=\frac{(\mathbf{a}+\mathbf{b})\,.\,\mathbf{b}}{|\mathbf{a}+\mathbf{b}|\,|\mathbf{b}|}$
• $\Rightarrow \frac{|\mathbf{a}{{|}^{2}}}{|\mathbf{a}+\mathbf{b}|\,|\mathbf{a}|}+\frac{\mathbf{b}\,.\,\mathbf{a}}{|\mathbf{a}+\mathbf{b}|\,|\mathbf{a}|}=\frac{\mathbf{a}\,.\,\mathbf{b}}{|\mathbf{a}+\mathbf{b}|\,|\mathbf{b}|}+\frac{|\mathbf{b}{{|}^{2}}}{|\mathbf{a}+\mathbf{b}|\,|\mathbf{b}|}$                    $\Rightarrow \frac{|\mathbf{a}|-|\mathbf{b}|}{|\mathbf{a}+\mathbf{b}|}\left( 1-\frac{\mathbf{a}\,.\,\mathbf{b}}{|\mathbf{a}|\,|\mathbf{b}|} \right)=0$
• Hence $|\mathbf{a}|\,=\,|\mathbf{b}|$ or angle between $\mathbf{a}$ and $\mathbf{b}$ is 0.