Exercise 8

This exercise introduces a version of the Pythagorean theorem that we'll use later.

1. Suppose that $\vvec$ and $\wvec$ are orthogonal to one another. Use the dot product to explain why $$\len{\vvec+\wvec}^2 = \len{\vvec}^2 + \len{\wvec}^2.$$

2. Suppose that $W$ is a subspace of $\real^m$ and that $\zvec$ is a vector in $\real^m$ for which $$\zvec = \xvec + \yvec,$$ where $\xvec$ is in $W$ and $\yvec$ is in $W^\perp$. Explain why $$\len{\zvec}^2 = \len{\xvec}^2 + \len{\yvec}^2,$$ which is an expression of the Pythagorean theorem.