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| },

and

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!

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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:

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

X=rE.

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

\frac{V}{|V|}.

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

\frac{V}{|V|}.

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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