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 byThen 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