Question

This question is from my linear algebra class:

Suppose u and w are unital orthogonal vectors. Show that || u - w || =
`√(2)`

Answer 1

Explanation:

Since u and w are orthogonal, their dot product is 0. Thus,

u · w = 0

Using the Pythagorean Theorem,

||u - w||^2 = ||u||^2 + ||w||^2 - 2(u · w)

Since u and w are unital (i.e., their lengths are 1), we have

||u||^2 = ||w||^2 = 1

Substituting these values and u · w = 0, we get

||u - w||^2 = 1 + 1 - 0 = 2

Taking the square root of both sides, we get

||u - w|| = √2

Therefore, we have shown that if u and w are unital orthogonal vectors, then ||u - w|| = √2.