Relation between position vectors of a rectangle

Question

I am given the position vectors of the rectangle a,b,c,d.

I am supposed to prove that a.c=b.d (.=dot product)

I tried representing the adjacent sides in terms of a,b,c,d since their dot product is 0(adjacent sides are perpendicular). But many big equations came that gave no solution. I even tried equating cross product of opposite sides is 0 but to no avail.Please guide me in the right direction since i thought of everything i could?

I suspect it has something to do with the geometrical meaning of dot product

Answer 1

You can use the above diagram.

Calling the four vertices as $$ \vec a=(a,0)\quad \vec b=(a,x)\,,\quad \vec c=(c,0)\,,\quad \vec d=(c,x)$$ where $x$ is the height of the rectangle, we can calculate the desired dot products. For example, $\vec b \cdot \vec c=ac$ and $\vec a \cdot \vec d=ac$.

In such problems you can always choose a coordinate system that make the problem as simple as possible. It is not necessary to consider the rectangle in the middle of page, since this problem is independent of the frame.

Answer 2

Choose one of these vectors — let's say, $a$, — and express the rest of the vectors through $a$ and the sides of the rectangle.