Linear Algebra. Help!!?

First, note that (1, 3) and (-2, 1) are orthogonal with respect to this 'new' inner product, since <(1, 3), (-2, 1)> = 3 * 1 * -2 + 2 * 3 * 1 = 0.

Now, we normalize these vectors with respect to this inner product.
||(1, 3)|| = √(<(1, 3), (1, 3)>) = √(3 * 1 * 1 + 2 * 3 * 3) = √21
||(-2, 1)|| = √(<(-2, 1), (-2, 1)>) = √(3 * -2 * -2 + 2 * 1 * 1) = √14.

Then, (1, 3)/√21 = (1/√21, 3/√21) and (-2, 1)/√14 = (-2/√14, 1/√14) are orthogonal with respect to the 'new' inner product.

However, (1/√21, 3/√21) and (-2/√14, 1/√14) are not orthogonal with respect to the usual Euclidean inner product, because their Euclidean inner product equals (1 * -2 + 3 * 1)/(√21 √14) = 1/(7√6), which is nonzero.

I hope this helps!