MAT224H1 Lecture : Trick that allows expression of angles
Document Summary
We started with a digression, describing a trick that allows to express angles in terms of distances (a sort of cosine law): Problem let t : v v be an operator satisfying |t v| = |v| for all v v . Prove that t v t w = v w for all v, w v . The meaning of this problem is that an operator that preserves distances must also preserve dot products and hence angles (v w = |v||w| cos 6 v, w). Solution we will use the identity v w = (|v +w|2 |v w|2)/4 expressing dot products in terms of distances. The proof of this identity is a straight- forward calculation: |v + w|2 = (v + w) (v + w) = |v|2 + |w|2 + 2v w. |v w|2 = (v w) (v w) = |v|2 + |w|2 2v w. Subtracting these equations we get the desired result. = |t (v + w)|2 |t (v w)|2.
