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If $|\vec{a}|=2,|\vec{b}|=5$ and $|\vec{a} \times \vec{b}|=8$, then what is the value of
$\vec{a} \cdot \vec{b} ?$
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$\vec{a} \cdot \vec{b} ?$
Solution:
1356 Upvotes
Verified Answer
The correct answer is:
6
We know that
$|\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}|^{2}+|\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}|^{2}=|\overrightarrow{\mathbf{a}}|^{2} \times|\overrightarrow{\mathbf{b}}|^{2}$
$\therefore \quad 64+|\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}|^{2}=(4 \times 25)$
$\Rightarrow|\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}|^{2}=36$
$\Rightarrow \overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=6$
$|\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}|^{2}+|\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}|^{2}=|\overrightarrow{\mathbf{a}}|^{2} \times|\overrightarrow{\mathbf{b}}|^{2}$
$\therefore \quad 64+|\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}|^{2}=(4 \times 25)$
$\Rightarrow|\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}|^{2}=36$
$\Rightarrow \overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=6$
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