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If $\vec{a}$ and $\overrightarrow{\mathrm{b}}$ are vectors such that $|\vec{a}|=2,|\overrightarrow{\mathrm{b}}|=7$ and
$\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+6 \hat{\mathrm{k}}$, then what is the acute angle between
$\vec{a}$ and $\overrightarrow{\mathrm{b}}$ ?
Options:
$\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+6 \hat{\mathrm{k}}$, then what is the acute angle between
$\vec{a}$ and $\overrightarrow{\mathrm{b}}$ ?
Solution:
1633 Upvotes
Verified Answer
The correct answer is:
$30^{\circ}$
$\begin{aligned} &|\overrightarrow{\mathrm{a}}|=2,|\overrightarrow{\mathrm{b}}|=7 \\ & \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+6 \hat{\mathrm{k}} \end{aligned}$
We know, $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}| \sin \theta \cdot \hat{\mathrm{n}}$ where $\hat{\mathrm{n}}$ is unit
vector.
$\Rightarrow|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|=|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}| \sin \theta$
$\Rightarrow|3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+6 \hat{\mathrm{k}}|=(2)(7) \sin \theta$
$\Rightarrow \sqrt{9+4+36}=(2)(7) \sin \theta .$
$\Rightarrow \pm 7=(2)(7) \sin \theta .$
$\therefore \sin \theta=\pm \frac{1}{2} .$
$\sin \theta$ is acute angle, $\theta=30^{\circ}$
We know, $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}| \sin \theta \cdot \hat{\mathrm{n}}$ where $\hat{\mathrm{n}}$ is unit
vector.
$\Rightarrow|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|=|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}| \sin \theta$
$\Rightarrow|3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+6 \hat{\mathrm{k}}|=(2)(7) \sin \theta$
$\Rightarrow \sqrt{9+4+36}=(2)(7) \sin \theta .$
$\Rightarrow \pm 7=(2)(7) \sin \theta .$
$\therefore \sin \theta=\pm \frac{1}{2} .$
$\sin \theta$ is acute angle, $\theta=30^{\circ}$
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