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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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4 Answers

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Two vectors $v_1=(x_1,y_1)$ and $v_2=(x_2,y_2)$ are parallel iff $x_1 \, y_2 = x_2 \, y_1$.

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You can find the angle between the two vectors $$\theta=\cos^{-1}(\frac{v_1.v_2}{|v_1||v_2|})$$ if $\theta=0$or $180$ the two vectors are parallel

if $\theta=90$ the two vetors are perpendicular

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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).

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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.

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