Question

If $|a|=8,|b|=3$ and $|a\times b|=12$, then the value of $a\cdot b$ is

• A. $6$ or $-6$
• B. $12\sqrt{3}$ or $-12\sqrt{3}$
• C. $8$ or $-8$
• D. None of these

Answer 1

The correct option is B $12\sqrt{3}$ or $-12\sqrt{3}$

$|a\times b|=12$

$\left|a\right|\left|b\right|\sin \theta =12$

$\sin \theta =\frac{12}{\left|a\right|\left|b\right|}$

$\sin \theta =\frac{12}{8\times 3}$

$\sin \theta =\frac{1}{2}$

$\theta ={30}^{\circ }$ or ${150}^{\circ }$

$a.b=\left|a\right|\left|b\right|\cos \theta$

$a.b=8\times 3\times \cos 30$

$a.b=8\times 3\times \frac{\sqrt{3}}{2}$

$a.b=12\sqrt{3}$

Similarly,

$a.b=\left|a\right|\left|b\right|\cos \theta$

$a.b=8\times 3\times \cos 150$

$a.b=-8\times 3\times \frac{\sqrt{3}}{2}$

$a.b=-12\sqrt{3}$