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Prove statements about a vector satisfying a given vector equation

Consider unit length, orthogonal vectors , and a vector such that Prove the following.

1. and are orthogonal and the length of is .
2. The vectors form a basis for .
3. .
4. .

1. 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 2. 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 3. 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 4. Proof. We compute One comment

1. Eiji says:

For 15(a),you calculated ∣∣A-P∣∣² wrong. ∣∣A-P∣∣² should be as follows:
∣∣A-P∣∣² = ∣∣A∣∣² – 2A⋅P + ∣∣P∣∣².

Then you should get ∣∣A∣∣² – 2A⋅P + ∣∣P∣∣² = ∣∣P∣∣². So, A⋅P = ½.

Now you can go to the original equation PXB = A-P, dot product both sides by P and get P⋅(PXB) = P⋅(A-P). (=>) 0 = P.A – P⋅P. (=>) ½ = P⋅P. And you have P = √2/2