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

Therefore,

- Again, we start with definition of ,

But, from part (b) we know . Hence, .

*Related*