I'm not sure if I quite get this. For example,
(1, -1) and (-3, 3)
take the cross product, you will end up with -3 + (-3)
This doesn't equal 0, so it's not perpendicular. So that leaves me with it being parallel. When are two vectors parallel?
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Two vectors are parallel when they are scalar multiples of each other. In other words, if you can multiply one vector by a constant and end up with the other vector.
The rough reason for this is that multiplying by a scalar doesn't rotate the vector at all (it can stretch or flip the vector, but it doesn't change the direction).
They are parallel if and only if they are different by a factor i.e. (1,3) and (-2,-6). The dot product will be 0 for perpendicular vectors i.e. they cross at exactly 90 degrees.
When you calculate the dot product and your answer is non-zero it just means the two vectors are not perpendicular.