Dear Professor.

I want to ask you a maths question. Hope you don’t mind.

What is the difference between

and

?

Where is the projection of on .

I understand that the first expression is the magnitude of , which is the length of the projection, am i right?

However, I’m uncertain about the second expression and to what significance it holds. Please enlighten me regarding this.

Thank you very much! Your help is much appreciated.

Take care!

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Reply:

You are right about the length of the projection. Perhaps the best way to see the meaning of the second expression is to write it as:

Here is the point: Given any vector , I believe you know how to visualize its projection along any directed line . For simplicity, I will assume that forms an acute angle with the direction of the line. The projection is then the shadow of you see when you shine a light on along a perpendicular angle. It is an elementary exercise with a triangle to see that the length of this projection is exactly

where is a *unit vector*, that is, a vector of length 1, in the direction of . You should think of a unit vector as encoding ‘pure direction.’ So to get a vector of magnitude in the direction of the unit vector , you can just dilate by a factor of to get

Conversely, to change any non-zero vector into the unit vector in the same direction, you divide by its magnitude to get

Note now the key fact that the projection of itself, not just its magnitude, is given by the formula

since the projection is clearly in the direction of and this expression gives such a vector with just the right magnitude.

To return to your question, given any two vectors and , you can project in the direction of . Now, the unit vector in the direction of is exactly

So the projection of in the direction of is nothing but

By the way, your question is a very good one. It’s very important to understand the geometric meaning of basic expressions like this. Otherwise, everything degenerates into a bunch of formulas.

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