find scalar product of vectors in rectangular

Question

let us consider following problem and picture

we have $ABCD$ rectagular with $AB=3$ and $BC=5$,$F$ and $E$ are midpoints of rectangular sides,we should find scalar product of

my question is can i locate point $A$ arbitrary or could i take coordinates of points arbitrary so that satisfy length property,namely $AB=3$,$BC=5$ or?clearly it seems that angle $FEC$ are not orthogonal,but how can i proof it?

Answer 1

By Chasles relation we have $$\vec{EF}\cdot\vec{EC}=(\vec{EA}+\vec{AF})\cdot(\vec{ED}+\vec{DC})=\vec{EA}\cdot\vec{ED}+\vec{AF}\cdot\vec{DC}\\ =-\frac{1}{4}(BC)^2+\frac{1}{2}(AB)^2=-\frac{7}{4}$$

Answer 2

Yes, if point $E$ and $F$ and $C$ are translated by $X$ then the vectors are $(F+X)-(E+X)=F-E$ and $(C+X)-(E+X)=C-E$, the same as if no translation.

Even more, you can also apply any rotation. They are transformations that preserve the scalar product.