Mathematics - Vector Question with Solution | TestHub

$\left|\overrightarrow{AB}\right|=a$

$\left|\overrightarrow{AD}\right|=b$

$\left|\overrightarrow{AC}\right|=\left|\overrightarrow{b}+\overrightarrow{a}\right|=c$

Squaring both side ${\left|\overrightarrow{b}\right|}^2+{\left|\overrightarrow{a}\right|}^2-2\overrightarrow{b}·\overrightarrow{a}=c^2$

$b^2+a^2-c^2=2\overrightarrow{b}·\overrightarrow{a}\dots \left(1\right)$

$\overrightarrow{DB}.\overrightarrow{AB}=\left(\overrightarrow{a}-\overrightarrow{b}\right)·\left(\overrightarrow{a}\right)$

$={\left(\overrightarrow{a}\right)}^2-\overrightarrow{a}·\overrightarrow{b}$

$=\frac{1}{2}(a^2-b^2+c^2)$