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## Homework Statement

Looking for some help with the proof if possible.

Vector r =

x

y

z

Rotation R =

cos(θ) 0 sin(θ)

0 1 0

-sin(θ) 0 cos(θ)

r' = Rr

It asks me to prove that

r'.r' = r.r

Second part of the question is about eigenvalues, it asks me to find the three eigenvalues of R.

I used the formula

det(m - λI)

Where m = R, λ = the eigenvalues and I is the appropriate identity matrix

I end up with λ = cos() or 1, which is clearly wrong as there isn't enough answers.

Somebody in class mentioned imaginary numbers, but I'm unsure as to how to proceed.

## The Attempt at a Solution

First Part:

I found r' to be

xcos(θ) + xsinθ

y

-zsin(θ) + xcos(θ)

To get r'.r' am I right to just multiply two of the above together, as in

(xcos(θ) + xsin(θ))(xcos(θ) + xsin(θ))

(yy)

(-zsin(θ) + xcos(θ))(-zsin(θ) + xcos(θ))

Because this is the way I did it and it doesn't lead to the same answer.

Obviously these should all have big brackets around them, but I am unsure of how to represent them on here, if someone would advise me I would gladly fix that.

Second Part:

I used the formula

det(m - λI)

Where m = R, λ = the eigenvalues and I is the appropriate identity matrix

I end up with λ = cos() or 1, which is clearly wrong as there isn't enough answers.

Somebody in class mentioned imaginary numbers, but I'm unsure as to how to proceed.