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Are two vectors in the same direction if their dot product is greater than zero/positive? I know they are orthogonal if their dot product is 0 so they can not be in the same direction. I also read if a vector u is scalar multiple of v, they are in the same direction? I can not find a definitive answer.

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  • $\begingroup$ One should be a positive scalar multiple of the other. $\endgroup$
    – Randall
    Commented Nov 5, 2017 at 3:07
  • $\begingroup$ @Randall what about if their dot product is positive? Can we say the two vectors are in the same direction and opposite direction if negative? $\endgroup$ Commented Nov 5, 2017 at 3:08
  • $\begingroup$ Here’s the MathJax tutorial $\endgroup$ Commented Nov 5, 2017 at 3:09
  • $\begingroup$ Positive dot product means acute angle, not the same direction necessarily. $\endgroup$
    – Randall
    Commented Nov 5, 2017 at 3:10
  • $\begingroup$ Consider $(0,1)$ and $(1,1)$ (i.e. east and northeast) I would not say are in the same direction, this is despite their dot product $(0,1)\cdot (1,1)=0\cdot1+1\cdot1=1>0$. All that you can say when the dotproduct is nonzero is that they are not orthogonal. $\endgroup$
    – JMoravitz
    Commented Nov 5, 2017 at 3:11

3 Answers 3

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Two vectors are in exactly the same direction if one is a positive scalar multiple of the other. Related facts:

  • Two vectors form an acute angle if their dot product is positive, and
  • two vectors form an obtuse angle if their dot product is negative.
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visual

Two vectors $\mathbf v$ and $\mathbf w$ are in the same direction if and only if $$\frac{\mathbf{v}}{v}\cdot\frac{\mathbf{w}}{w}=1$$

One of the many ways your can rephrase this is $\mathbf{\hat v}=\mathbf{\hat w}$. You are right that they are scalar multiples.

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  • $\begingroup$ $v$ and $w$ are lengths of $\mathbf v$ and $\mathbf w$, right? $\endgroup$
    – Cm7F7Bb
    Commented Oct 27, 2022 at 14:07
  • $\begingroup$ Being scalar multiples is necessary but not sufficient, since for $v \neq 0, v$ and $-v$ are scalar multiples but not in the same direction. $\endgroup$ Commented Dec 28, 2022 at 23:51
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Vectors u and v are in same direction if their unit norm are equal ie vectors are scalar multiple of each other. $$\frac{u}{||u||}=\frac{v}{||v||}$$

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  • $\begingroup$ I believe this incorrect, since a. they need to be positive scalar multiples and b. the zero vector is in the same direction as itself even though it does not have a unit norm $\endgroup$ Commented Dec 28, 2022 at 23:52

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