Question

If $|\overrightarrow{a}|=5,|\overrightarrow{b}|=6$ and $\overrightarrow{a}.\overrightarrow{b}=-25,$ then $|\overrightarrow{a}\times \overrightarrow{b}|$ is equal to

• A. 25
• B. $6\sqrt{11}$
• C. $11\sqrt{5}$
• D. $11\sqrt{6}$
• E. $5\sqrt{11}$

Answer 1

Answer: (D)

Given, $|\overrightarrow{a}|=5,|\overrightarrow{b}|=6,\overrightarrow{a}.\overrightarrow{b}=-25$
$\therefore$ $\cos \theta =\frac{\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{a}|.|\overrightarrow{b}|}=\frac{-25}{5\times 6}=-\frac{25}{30}=-\frac{5}{6}$
$\therefore$ $\sin \theta =\sqrt{1-{\cos }^2\theta }=\sqrt{1-\frac{25}{36}}$
$=\sqrt{\frac{11}{36}}$
$\therefore$ $|\overrightarrow{a}\times \overrightarrow{b}|=|\overrightarrow{a}||\overrightarrow{b}|\sin \theta |$
$=5\times 6\times \sqrt{\frac{11}{36}}$
$=5\sqrt{11}$