Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I'm reading Chris Hecker's third article on rigid body dynamics


"More importantly, if our collision detector supplies us with a 'normal vector' for the collision (denoted by n, and pointing toward body A by convention), we can define the 'relative normal velocity' as the component of the relative velocity in the direction of the collision normal."

Which he defines as vAB . n where vab is the relative velocity of points A and B and n is the normal vector for the collision.

I read Understanding Dot and Cross Product which explains that the dot product gives the length of one vector in the direction of another, which I think is what is being applied here, but I'm having a really hard time visualizing what is going on, specifically what the component is.

Can anyone help explain what this component is and how using the dot product helps identify it?

share|cite|improve this question
This Wikipedia article might help. – Jonas Meyer Jun 14 '11 at 5:45
up vote 1 down vote accepted

If you have vectors ${\bf x}$ and ${\bf y}$, you can write ${\bf x}={\bf u}+{\bf v}$, where ${\bf u}$ is parallel ("in the direction of") ${\bf y}$ and ${\bf v}$ is perpendicular to ${\bf y}$. Visually, drop a perpendicular from ${\bf x}$ onto ${\bf y}$ (first extending $\bf y$, if necessary), then that perpendicular is $\bf v$ and the vector from the origin to the foot of the perpendicular is $\bf u$.

Now from ${\bf x}={\bf u}+{\bf v}$ you get ${\bf x}\cdot{\bf u}={\bf u}\cdot{\bf u}+{\bf u}\cdot{\bf v}$. Now ${\bf u}\cdot{\bf v}=0$ since $\bf u$ and $\bf v$ are perpendicular, and ${\bf u}\cdot{\bf u}$ is the square of the length of $\bf u$, so ${\bf x}\cdot{\bf u}=\|{\bf u}\|^2$. That gives you the interpretation of the dot product in terms of the component of one vector in the direction of another.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.