r"(t) x r'(t) = 0. prove that r(t) moves in a line
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if the cross product of two vector equations is zero, they are either the same non-zero vector, or one of them has components (or magnitude) of zero. In this case, they cannot be the same because one is the derivative of the other, so one (or both) must be a zero vector. r''(t) must be a zero vector because it is the second order derivative of r(t) and r'(t) is only the first order derivative. So if r''(t) is zero, then r'(t) must have components which are constants (e.g. <1,2,3>), and r(t) will have components which progress linearly (e.g. <1t,2t,3t>) so r(t) must be linear. if both r''(t) and r'(t) are zero vectors then r(t) will have components which are constants (<1,2,3>) which is still a line.