Question

Calculate the scalar product of the following vectors.
Given a parallelogram ABCD. The lengths of the vectors $\overrightarrow{AB},\overrightarrow{AD}and\overrightarrow{BD}$ are a, b and c respectively. Find the scalar product of the vectors $\overrightarrow{AC}and\overrightarrow{AD}.$

Answer 1

we know that the diagonals of a parallelogram have the same length. Thus $|\overrightarrow{AC}|=|\overrightarrow{BD}|=c$

using cos rule, $\cos \angle DAC=\frac{b^2+c^2-a^2}{2bc}$

$⟹\sin \angle DAC=\sqrt{\frac{(2bc{)}^2-(b^2+c^2-a^2{)}^2}{(2bc{)}^2}}=\frac{\sqrt{2a^2{c}^2+2a^2{b}^2+2b^2{c}^2-a^4-b^4-c^4}}{2bc}$

$\overrightarrow{AC}.\overrightarrow{AD}=bc\sin \angle DAC=bc\sqrt{\frac{(2bc{)}^2-(b^2+c^2-a^2{)}^2}{(2bc{)}^2}}=\frac{\sqrt{2a^2{c}^2+2a^2{b}^2+2b^2{c}^2-a^4-b^4-c^4}}{2}$