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