Consider unit length, orthogonal vectors , and a vector such that
Prove the following.
- and are orthogonal and the length of is .
- The vectors form a basis for .
- Proof. We compute,
since and since and are orthogonal by assumption. Thus, and are orthogonal. Next,
since by hypothesis and . Hence, from the vector equation we have
- Proof. Since and are orthogonal (part (a)), we know the vectors are independent. Thus, they form a basis for since any three independent vectors in are a basis
- Proof. We compute, the vector is given by
Then the three coordinates of this cross product are given by
Expanding these out we obtain the coordinates
Since we know and since we know . So, simplifying the expressions above, for each of the coordinates we have
Hence, we indeed have
- Proof. We compute