If ${\overrightarrow{\mathbf{u}}}_{1}\ and\ {\overrightarrow{\mathbf{u}}}_{2}$ be vectors of unit length and $ heta$ be the angle between them, then $\frac{1}{2}|{\overrightarrow{\mathbf{u}}}_{2} - {\overrightarrow{\mathbf{u}}}_{1}|is$

Question

If ${\overrightarrow{\mathbf{u}}}_{1}\ and\ {\overrightarrow{\mathbf{u}}}_{2}$ be vectors of unit length and $	heta$ be the angle between them, then $\frac{1}{2}|{\overrightarrow{\mathbf{u}}}_{2} - {\overrightarrow{\mathbf{u}}}_{1}|is$

Dr. Abubacker · Accepted Answer

(b)
$\left\lbrack \frac{1}{2}\left| {\overrightarrow{\mathbf{u}}}_{2} - {\overrightarrow{\mathbf{u}}}_{1} ight| ightbrack^{- 2} = \frac{1}{4}\lbrack|{\overrightarrow{\mathbf{u}}}_{2}|^{2} + |{\overrightarrow{\mathbf{u}}}_{1}|^{2} - 2{\overrightarrow{\mathbf{u}}}_{2} \bullet {\overrightarrow{\mathbf{u}}}_{1}brack$
$= \frac{1}{4}\lbrack 1 + 1 - 2\left| {\overrightarrow{\mathbf{u}}}_{2} ight|\left| {\overrightarrow{\mathbf{u}}}_{1} ight|\cos{	hetabrack}$
$= \frac{1}{4}\lbrack 2 - 2\cos{	hetabrack = \sin^{2}\frac{	heta}{2}}$
$\Longrightarrow \frac{1}{2}\left| {\overrightarrow{\mathbf{u}}}_{2} - {\overrightarrow{\mathbf{u}}}_{1} ight| = \sin\frac{	heta}{2}$

Q. If ${\overrightarrow{\mathbf{u}}}_{1}\ and\ {\overrightarrow{\mathbf{u}}}_{2}$ be vectors of unit length and $\theta$ be the angle between them, then $\frac{1}{2}|{\overrightarrow{\mathbf{u}}}_{2} - {\overrightarrow{\mathbf{u}}}_{1}|is$

Explanation

Reviewed by Dr. Abubacker

Want more practice?