In a parallelogram ABCD, AB=a, AD=b and AC=c, then DAAB has the value:

Question

In a parallelogram $ABCD$, $\overrightarrow{AB}=a$, $\overrightarrow{AD}=b$ and $\overrightarrow{AC}=c$, then $\overrightarrow{DA}\cdot\overrightarrow{AB}$ has the value:

Accepted Answer

$\overrightarrow{AB}+\overrightarrow{AD}=\overrightarrow{AC}$. Squaring: $|\overrightarrow{AB}|^2+|\overrightarrow{AD}|^2+2\overrightarrow{AB}\cdot\overrightarrow{AD}=|\overrightarrow{AC}|^2$

$$a^2+b^2+2\overrightarrow{AB}\cdot(-\overrightarrow{DA})=c^2 \implies \overrightarrow{DA}\cdot\overrightarrow{AB}=\frac12(a^2+b^2-c^2)$$

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