Prove that there is one and only one vector satisfying the equations

where and is orthogonal to in .

*Proof.* First, we show uniqueness. If is any vector such that and and is another vector such that and then we have

But, by a previous exercise (Section 13.11, Exercise #8) we know these conditions imply . Hence, there is at most one such .

Now, for existence. Let . Since are orthogonal we know ,

From we then have

Since , we know at least one of the . Without loss of generality, assume . From the second and third equations we have

This implies

Hence, can take any value. The vectors such that are then of the form

Then,

Since (since at least one of the , we have that such a vector always exists

There is a calculation error. The equation a2b3-a3b2=c1 should lead to a2a3-a3a2.