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
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
Wrong problem ignore what I said ._.
There is a calculation error. The equation a2b3-a3b2=c1 should lead to a2a3-a3a2.