# What separates the dot product from the scalar projection?

Just a little problem with geometric intuition here (or perhaps I just haven't slept in far too long!).

I know that the scalar projection of vectors $\vec{u}$ and $\vec{v}$ is defined as

$\displaystyle proj_{\vec{v}}\vec{u} = \frac{\vec{u} \cdot \vec{v}}{\| \vec{v} \|}$

I read in another thread that the dot product represents the "amount one vector goes in the direction of another". However, I thought that was the definition of the scalar projection?

They are obviously not the same thing (however I can see that the scalar projection is derived from the geometric definition of the dot product); the scalar projection is the dot product divided by the magnitude of one of the vectors:

$\displaystyle \vec{u} \cdot \vec{v} = \| \vec{u} \| \| \vec{v} \| \cos{\theta}$

$\displaystyle \frac{\vec{u} \cdot \vec{v}}{\| \vec{v} \|} = \| \vec{u} \| \cos{\theta} = proj_{\vec{v}}\vec{u}$

What am I missing, in general, about the differences or similarities between these two things?

## 1 Answer

• Dot project is symmetric: $\vec{u} \cdot \vec{v}=\vec{v} \cdot \vec{u}$. Scalar projection is not.

• Dot project is sensitive to the magnitude of the second vector : $\vec{u} \cdot k\vec{v}=k(\vec{u} \cdot \vec{v})$. Scalar projection is not.

• If your distances are measured in metres then the dot product is measured in square metres but the scalar projection is measured in metres.