# Two vectors are in the same direction if?

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.

• One should be a positive scalar multiple of the other. Nov 5, 2017 at 3:07
• @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? Nov 5, 2017 at 3:08
• Here’s the MathJax tutorial Nov 5, 2017 at 3:09
• Positive dot product means acute angle, not the same direction necessarily. Nov 5, 2017 at 3:10
• 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. Nov 5, 2017 at 3:11

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. 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.

• $v$ and $w$ are lengths of $\mathbf v$ and $\mathbf w$, right? Oct 27, 2022 at 14:07
• 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. Dec 28, 2022 at 23:51

Vectors u and v are in same direction if their unit norm are equal ie vectors are scalar multiple of each other. $$\frac{u}{||u||}=\frac{v}{||v||}$$

• 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 Dec 28, 2022 at 23:52