- #1

joshmccraney

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## Homework Statement

Show that the matrix ##P = \big{[} p_{ij} \big{]}## is orthogonal.

## Homework Equations

##P \vec{v} = \vec{v}'## where each vector is in ##\mathbb{R}^3## and ##P## is a ##3 \times 3## matrix. SO I guess ##P## is a transformation matrix taking ##\vec{v}## to ##\vec{v}'##. I also know ##\vec{v} = v_i \hat{e}_i## where ##\hat{e}_i## is the ##i##th unit vector.

## The Attempt at a Solution

Orthogonal implies ##P P^t = I##. ##P P^t## can be wrote in component form as ##p_{ij} p_{ji}##. I believe I want to show that ##p_{ij} p_{ji} = \delta_{ij}##. After this I'm not really sure how to proceed. Any ideas?