## 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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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.