Question

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two non-collinear unit vectors. If $\overrightarrow{u}=\overrightarrow{a}-(\overrightarrow{a}\cdot \overrightarrow{b})\overrightarrow{b}$ and $\overrightarrow{v}=\overrightarrow{a}\times \overrightarrow{b}$, then $|\overrightarrow{v}|$ is

• A. $|\vec{u}|$
• B. $|\vec{u}|+|\vec{u}.\vec{a}|$
• C. $|\vec{u}|+|\vec{u}.\vec{b}|$
• D. $|\vec{u}|+|\vec{u}.(\vec{b}+\vec{b})$

Answer 1

Answer: (A)

$|\vec{v}|=|\vec{a}\times \vec{b}|=|\vec{a}||\vec{b}|\sin \theta =\sin \theta$
$|\vec{u}|=|\vec{a}-(\vec{a}\cdot \vec{b})\vec{b}|=\sqrt{a^2+(\vec{a}\cdot \vec{b}{)}^2{b}^2-2(\vec{a}\cdot \vec{b}{)}^2}$
$=\sqrt{1+{\cos }^2\theta -2{\cos }^2\theta }$
$=\sqrt{1-{\cos }^2\theta }$
$=\sqrt{{\sin }^2\theta }$
$=\sin \theta$