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Home  >>  CBSE XII  >>  Math  >>  Vector Algebra

The vector $\overrightarrow{a}+\overrightarrow{b}$ bisects the angle between the non-collinear vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ if__________.

$\begin{array}{1 1} |\overrightarrow a|=|\overrightarrow b| \\ |\overrightarrow a|=2|\overrightarrow b| \\ 2 |\overrightarrow a|=|\overrightarrow b| \\ If \;\overrightarrow a\; and \;\overrightarrow b\; are\; \perp\end{array} $

1 Answer

Toolbox:
  • $\cos \theta=\large\frac{\overrightarrow a.\overrightarrow b}{|\overrightarrow a||\overrightarrow b|}$
Let us consider two non-collinear vectors $\overrightarrow a$ and $\overrightarrow b$
Let $\overrightarrow a+\overrightarrow b$ be the vector which bisects the angle between the two vectors.
Hence $ \theta_1=\theta_2$
Therefore $ \cos \theta_1=\large\frac{\overrightarrow a.(\overrightarrow a+\overrightarrow b)}{|\overrightarrow a||\overrightarrow a+\overrightarrow b|}$
$ \cos \theta_2=\large\frac{\overrightarrow b.(\overrightarrow a+\overrightarrow b)}{|\overrightarrow b||\overrightarrow a+\overrightarrow b|}$
Since $\theta_1=\theta_2$=> $\cos \theta_1=\cos \theta_2$
$ \large\frac{\overrightarrow a.(\overrightarrow a+\overrightarrow b)}{|\overrightarrow a||\overrightarrow a+\overrightarrow b|}=\large\frac{\overrightarrow b.(\overrightarrow a+\overrightarrow b)}{|\overrightarrow b||\overrightarrow a+\overrightarrow b|}$
$=>\overrightarrow a=\overrightarrow b$
The vector $\overrightarrow{a}+\overrightarrow{b}$ bisects the angle between the non-collinear vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are equal vectors
answered Jun 3, 2013 by meena.p
 

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