Proving Geometric Fact: u+v Perpendicular to u-v Using Dot Product

[Teggles]

"If u and v are any two vectors of the same length, use the dot product to show that
u + v is perpendicular to u − v. What fact from geometry is does this represent."

This is basically the last question in an assignment on vectors (first year university, linear algebra). The questions all focus on things like equations of planes, angles of intersection etc. that require little insight. This one however seems to require quite a lot of insight.

I understand how to calculate the dot product, cross product, length of vectors, angle between vectors, etc., but I don't even know where to start here.

If anyone could help me through this, it'd be extremely appreciated :)

 

[HallsofIvy]

Well, I hate to be obtuse but if you "understand how to calculate the dot product" and the problem tells you to use the dot product, isn't it obvious that you should start by taking the dot product?

What is the dot product $(u- v)\cdot(u+ v)[$?

 

[Teggles]

Well...

(u - v) . (u + v) = |u - v| |u + v| cos θ

Of course, I don't know |u - v| or |u + v|, only that |u| and |v| are the same. I don't know how this bit of information is meant to be used.

If the vectors are perpendicular, θ will be 90 degrees, and so dot product will be 0. Is my aim to therefore prove that the dot product is 0? How? Though the other method of calculating the dot product? I'm sorry, I just feel clueless.

 

[spamiam]

Try writing out the vectors' entries, i.e. u = (u₁, u₂, u₃) and v = (v₁, v₂, v₃) and use the entry-wise definition of the dot product. Also try writing out what |u| = |v| means in terms of the entries.

 

[HallsofIvy]

No! You don't have to be as complicated as either of those. $(u+ v)$$\cdot(u- v)$$= u\cdot u$$- u\cdot v$$+ v\cdot u- v\cdot v$$= |u|^2- |v|^2$ because the dot product is commutative so that $-u\cdot v+ v\cdot u= 0$

 

[Teggles]

Awesome. I managed to write a full proof that it would equal 0. I'm fairly glad you omitted some of the reasoning/detail because it actually made me think about and understand each step for myself. Thanks for your help!

Still haven't worked out what "geometric fact" the question wants, but I'll work on it.

 

[HallsofIvy]

Think about the "parallelogram" rule for adding vectors. If u and v are two sides of a parallelogram, what are u+ v and u- v?