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Say you are given theses two vectors:

u = (1, -2, 4)

v = (-2, 4, 8)

Since there are three entries, how do you know if they point in the opposite/same/different direction?

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1 Answer 1

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Regardless of the dimension, two vectors are in the same direction if the ratio between all pairs of corresponding components is the same and positive. They are opposite if the ratio is the same and negative. Otherwise they are in different directions. In your case, $\frac {-2}1=-2, \frac 4{-2}=-2,\frac 84=2$ Since they disagree, the vectors are in different directions. How could you change one component of one vector to make them opposite? How could you change two components of one vector to make them the same?

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