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This is an problem we have to do... I have done a lot of research on this. There is no help in our text book or previous homework. It gives: Show that a vector A(t) is parallel to its A"(t) for t is an element of [t1,t2], then A(t)xA'(t) is a constant vector field for t element of [t1,t2].

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    This should not require research. How do you always show some function is constant in the context of a calculus course? – 2017-01-30
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    if you show the first derivative is constant? – 2017-01-30
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    ... if you show the first deriviative is _zero._ (But perhaps that's what you really meant to say.) – 2017-01-30

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Hint 1: Recall (or look up) how to express the derivative of the cross product of two vectors in terms of the vectors and their derivatives.

Hint 2: Evaluate the derivative of $A(t)\times A'(t).$

(If your textbook has not yet covered the derivative of a cross product, you might be expected to prove the formula is correct. I hope that is not necessary.)