Question

If $\left|\overline{a}\right|=2,\left|\overline{b}\right|=5$ and $\left|\overline{a}\times \overline{b}\right|=8$, find $\overline{a}.\overline{b}$.

Answer 1

Given that

$|\overline{a}|=2,|\overline{b}|=5$

$|\overline{a}\times \overline{b}|=8$

Consider $|\overline{a}\times \overline{b}|=|\overline{a}|\cdot |\overline{b}|\cdot sin\theta$

$\Rightarrow 8=2\cdot 5sin\theta$

$\Rightarrow \frac{8}{10}=sin\theta$

$sin\theta =\frac{4}{5}$

$\theta =si{n}^{-1}\left(\frac{4}{5}\right)$

$\overline{a}\cdot \overline{b}=|\overline{a}|\cdot |\overline{b}|cos\theta$

$=2\cdot 5cos\left[si{n}^{-1}\left(\frac{4}{5}\right)\right]$

$=10cos\left[co{s}^{-1}\left(\frac{3}{5}\right)\right]$

$=10\times \frac{3}{5}=6$