Question

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

Answer 1

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

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

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

$\overrightarrow{DB}.\overrightarrow{AB}=ac\sin \angle DBA=ac\sqrt{\frac{(2bc{)}^2-(b^2+c^2-a^2{)}^2}{(2ac{)}^2}}=\frac{\sqrt{2a^2{c}^2+2a^2{b}^2+2b^2{c}^2-a^4-b^4-c^4}}{2}$