# Vector Rhombus Proof

## Homework Statement

Prove that the diagonals of a paraelogram are perpendicular iff the parallelogram is a rhombus.

a (dot) b = 0

## The Attempt at a Solution

This is how I started:

By definition, a rhombus is a quadrilateral with all sides equal in length. So this means that if I have two vectors, a and b that form the corner of a rhombus, then that means that the magintude of a and b are equal. By inspection of a diagram of this vector problem, I found that (a+b) (dot) (a-b) = 0 iff the magintude of a and b are equal.

This is great, however, it will not fly because I cannot just say "by inspection of the diagram" right. How can I put this in words that will make my proof make sense?

Thanks

The diagonals of the parallelogram are precisely a+b and a-b. if you are talking about proving your equation above, multiply it out, keeping in mind:

$$(a+b)\cdot (a-b) = a\cdot a - b\cdot b + b\cdot a - a\cdot b$$

edit: fixed my mistake

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The diagonals of the parallelogram are precisely a+b and a-b. if you are talking about proving your equation above, multiply it out, keeping in mind:

$$(a+b)\cdot (a-b) = a\cdot a + b\cdot b + b\cdot a - a\cdot b$$

AWESOME!! Thank you so much for you help. That was a lot easier than I thought. So, after multiplying it out, I came up with a²-b²=0. So this is true iff a² = b². Thanks for your help!