Let be two linearly independent vectors in and let
- Prove that and are orthogonal.
- Prove that where is defined to be the angle between and .
- Compute the length of if and .
- Proof. We have
from Theorem 13.12(e) (page 483 of Apostol). Hence, and are orthogonal
- Proof. Starting with the definition of we have
But, , so . Therefore, . Hence, we have
- Again, we start with definition of ,
But, from part (b) we know . Hence, .