Question

Find the value of $\left|\overrightarrow{a}\times \overrightarrow{b}\right|,$ ,if $\left|\overrightarrow{a}\right|=10,\left|\overrightarrow{b}\right|=2$ and $\overrightarrow{a}.\overrightarrow{b}=12$

Answer 1

given that

$\left|\overrightarrow{a}\right|=10,\left|\overrightarrow{b}\right|=2$ and $\overrightarrow{a}.\overrightarrow{b}=2$

then

we know formula

$\overrightarrow{a}.\overrightarrow{b}=\left|\overrightarrow{a}\right|.\left|\overrightarrow{b}\right|cos\theta$

$12=10\times 2cos\theta$ $\{\because \overrightarrow{a}.\overrightarrow{b}=12\}$

$\frac{6}{10}=cos\theta$

$cos\theta =\frac{3}{5}$

$\theta =co{s}^{-1}\left(\frac{3}{5}\right)$

$sin\theta =\frac{4}{5}$ $\{\because si{n}^2\theta +co{s}^2\theta =1\}$

then

we know $\overrightarrow{a}\times \overrightarrow{b}=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|sin\theta$

$=10\times 2\times sin\theta$

$=10\times 2\times \frac{4}{5}$

$=2\times 2\times 4$

$\overrightarrow{a}\times \overrightarrow{b}=16$

$\overrightarrow{a}\times \overrightarrow{b}=16$