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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?

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  • 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.

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