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# Prove that there is exactly one vector satisfying given conditions

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 1. William C says:

A dot B = A dot C implies A dot (B-C) = 0 which implies C = B-cA for some c

Using the above expression for c with A x B = A x C shows that c must equal 0 and thus C = B

• William C says:

Wrong problem ignore what I said ._.

2. Van Gogh says:

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