Let x and y be two vectors from Rn. Show that x − y andx + y are orthogonal if and only if ∥x∥ = ∥y∥

Sure, let's go through this step by step.

Two vectors are orthogonal if their dot product is zero. So, we need to show that (x - y) and (x + y) are orthogonal, i.e., their dot product is zero, if and only if ∥x∥ = ∥y∥.

Step 1: Assume ∥x∥ = ∥y∥. We need to show that (x - y) and (x + y) are orthogonal.

The dot product of (x - y) and (x + y) is:

(x - y) . (x + y) = x.x + x.y - y.x - y.y = ∥x∥^2 - ∥y∥^2 = 0

since ∥x∥ = ∥y∥. So, (x - y) and (x + y) are orthogonal.

Step 2: Now, assume that (x - y) and (x + y) are orthogonal. We need to show that ∥x∥ = ∥y∥.

Since (x - y) and (x + y) are orthogonal, their dot product is zero:

(x - y) . (x + y) = ∥x∥^2 - ∥y∥^2 = 0

which implies that ∥x∥^2 = ∥y∥^2, and taking square roots gives ∥x∥ = ∥y∥.

So, we have shown that x − y and x + y are orthogonal if and only if ∥x∥ = ∥y∥.