I proved that ||a+b|| = ||a-b|| if and only if a.b = 0
However the next part says using this explain why a paralleogram is a rhombus if and only if its diagonals are at right angles..
I can't figure this out?
I proved that ||a+b|| = ||a-b|| if and only if a.b = 0
However the next part says using this explain why a paralleogram is a rhombus if and only if its diagonals are at right angles..
I can't figure this out?
This is a neat problem.
If $\displaystyle \vec{a}~\&~\vec{b}$ are consecutive sides of a parallelogram then $\displaystyle \vec{a}+\vec{b}~\&~\vec{a}-\vec{b}$ are its diagonals.
Now $\displaystyle (\vec{a}+\vec{b}) \cdot (\vec{a}-\vec{b})=\vec{a}\cdot \vec{a}-\vec{b}\cdot\vec{b} $.
So if the parallelogram is a rhombus then $\displaystyle \left\| \vec{a} \right\|=\left\| \vec{b} \right\| $.
What does that imply?
What is the converse?