# If $\mid\overrightarrow a\mid=10,\mid\overrightarrow b\mid=2$ and $\overrightarrow a.\overrightarrow b=12$,then value of $\mid\overrightarrow a\times\overrightarrow b\mid$ is

$(A)\;5\quad(B)\;10\quad(C)\;14\quad(D)\;16$

Toolbox:
• $\cos \theta =\large\frac{\overrightarrow a.\overrightarrow b}{|\overrightarrow a||\overrightarrow b|}$
Given $|\overrightarrow a|=10,|\overrightarrow b|=2$ and $\overrightarrow a.\overrightarrow b=12$
$\large\frac{\overrightarrow a.\overrightarrow b}{|\overrightarrow a||\overrightarrow b|}$$=\cos \theta On substituting the values we get, \cos \theta=\large\frac{12}{10 \times 2}=\frac{3}{5} If \cos \theta=\large\frac{3}{5} then, \sin \theta=\large\frac{4}{5} we know \sin \theta =\large\frac{|\overrightarrow a \times \overrightarrow b|}{|\overrightarrow a||\overrightarrow b|} On substituting the values we get, =>\large\frac{4}{5}=\frac{|\overrightarrow a \times \overrightarrow b|}{10 \times 2} =|\overrightarrow a \times \overrightarrow b|=\large\frac{20 \times 4}{5}$$=16$
Hence $D$ is the correct option