Ev(Unit Vector) and projection of a vector in a dot product

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SUMMARY

The discussion centers on the mathematical concept of vector projection, specifically the projection of vector u onto the unit vector ev derived from vector v. The formula w = (u.ev)ev is established, where w represents the projection of u onto ev. The confusion arises from the presence of the additional ev in the equation, which is clarified as necessary to define w as a vector parallel to ev. The scalar (u.ev) alone does not suffice to represent the vector w without the unit vector component.

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So my book says

Lets suppose,

We have two vector v and u

w=projection of u ev= unit vector θ=angle between the two

w=(u.ev)ev or w=( (u.v)/(v.v) )v

Now, the second equation is fairly easy to understand if we understand the first one because ev= v / |v|

What is bothering me is I have no idea why w=(u.ev)ev.

It would be more reasonable, as per me, if w=(u.ev)

But that is not the case.

Why is that extra ev lingering around in the equation.

ANY IDEAS?
 
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w is a vector parallel to ev. (u.ev) is a scalar (± length of w), so you need the entire expression to define the vector w.
 

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