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If $|\vec{a}|=10,|\overrightarrow{\mathrm{b}}|=2$ and $\overrightarrow{\mathrm{a}} \bullet \overrightarrow{\mathrm{b}}=12$, then what is the value of
$|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}| ?$
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$|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}| ?$
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Verified Answer
The correct answer is:
16
We have
$\cos \theta=\frac{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}}{|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}|}$ and $\sin \theta=\frac{|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|}{|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}|}$
Given: $|\overrightarrow{\mathrm{a}}|=10,|\overrightarrow{\mathrm{b}}|=2, \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=12$
$\therefore \cos \theta=\frac{12}{20}$ and $\sin \theta=\frac{|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|}{20}$
Now, By squaring and adding, we get $\sin ^{2} \theta+\cos ^{2} \theta=1$
$\Rightarrow \frac{|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|^{2}}{400}+\frac{144}{400}=1 \Rightarrow|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|^{2}=256$
$\Rightarrow|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|=16$
$\cos \theta=\frac{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}}{|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}|}$ and $\sin \theta=\frac{|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|}{|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}|}$
Given: $|\overrightarrow{\mathrm{a}}|=10,|\overrightarrow{\mathrm{b}}|=2, \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=12$
$\therefore \cos \theta=\frac{12}{20}$ and $\sin \theta=\frac{|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|}{20}$
Now, By squaring and adding, we get $\sin ^{2} \theta+\cos ^{2} \theta=1$
$\Rightarrow \frac{|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|^{2}}{400}+\frac{144}{400}=1 \Rightarrow|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|^{2}=256$
$\Rightarrow|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|=16$
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