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)

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