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  • Fill in the blanks The vector a + b bisects the angle between the non-collinear vectors a and b if ________

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The vector \vec{a} + \vec{b}  bisects the angle between the non-collinear vectors \vec{a}  and \vec{b}  if ________

 

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Let  \vec{a}  and \vec{b}  are two non-collinear vectors.

Let \vec{a} + \vec{b}  bisects the angle between  \vec{a}  and \vec{b} .

\\ \Rightarrow \theta_{1}=\theta_{2} \\ \qquad \cos \theta_{1}=\frac{\vec{a} \cdot(\vec{a}+\vec{b})}{|\vec{a}||\vec{a}+\bar{b}|} \text { and } \cos \theta_{2}=\frac{\vec{b} \cdot(\vec{a}+\vec{b})}{|\vec{b}||\vec{a}+\vec{b}|} \\ \text { since, } \theta_{1}=\theta_{2} \Rightarrow \cos \theta_{1}=\cos \theta_{2} \\ \therefore \quad \frac{\vec{a} \cdot(\vec{a}+\vec{b})}{|\vec{a}||\vec{a}+\vec{b}|}=\frac{\vec{b} \cdot(\vec{a}+\vec{b})}{|\vec{b} \| \vec{a}+\vec{b}|} \\ \Rightarrow |\vec{a}|=|\vec{b}|

Thus, the vector \vec{a} + \vec{b}  bisects the angle between the non-collinear vectors \vec{a}  and \vec{b}   if they are equal.

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