Do the head to tail method and the parallelogram method always produce the same answer? Why?

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brehms - Yes! The resultant vector will always be equal. If you use head to tail, then the resultant sum of two vectors is the vector from the tail of the first vector to the head of the second vector. That same resultant vector will always equal the "diagonal" of the parallelogram formed by the two separate vectors. Best way to convince yourself is to simply draw several 2-vector sums. First using head-tail and then overlay the parallelogram method to "see" that this will always to true. Hope that helps!