Vector question

Dear Professor.

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

What is the difference between

|C| = \frac {A \cdot B} {|B| },


C = \frac {A \cdot B} {|B|^2} B ?

Where C is the projection of A on B.

I understand that the first expression is the magnitude of C, 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!


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:

C=(A \cdot (\frac{B}{|B|}))\frac{B}{|B|}

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

W\cdot e_L,

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


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


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

(W\cdot e_L)e_L

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

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


So the projection of W in the direction of V is nothing but

(W\cdot (\frac{V}{|V|}))\frac{V}{|V|}.

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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