Understanding Scalar and Vector Projections: A Layman's Guide

[kylera]

I'm re-visiting calculus again, and I've stumbled onto the concepts of scalar and vector projections in the vector chapter. While keeping in mind which equation to use for what projection is quite doable, I cannot seem to see the purpose of keeping scalar and vector projections in mind. Can anyone help clarify or state these two things in layman's terms? Much thanks in advance.

 

[HallsofIvy]

Imagine a vector with its tail at (0,0,0) and extending up to (1, 1, 1). Now imagine a light shining uniformly down from the z direction. The "shadow" of the vector <1, 1, 1> is its "vector projection" on the xy-plane (and would be <1, 1, 0>)) The length of that vector, $\sqrt{2}$, would be its scalar projection. As to why you should "keep that in mind", it depends on your purpose. If you were and engineer, I can think many reasons why you would want to know that. If you were taking a physics or calculus III test you would surely want to know it! If your goal in life is to say "Do you want fries with that?", then you have no need to know it at all.

 

[kylera]

Imagine a vector with its tail at (0,0,0) and extending up to (1, 1, 1). Now imagine a light shining uniformly down from the z direction. The "shadow" of the vector <1, 1, 1> is its "vector projection" on the xy-plane (and would be <1, 1, 0>)) The length of that vector, $\sqrt{2}$, would be its scalar projection.

That's a very good way to associate with the term "projection". I wish the book could put it that succinctly. Much much thanks!