Can 2 times the dot product of 2 complex numbers equal to this?

There's a section in Gibson's "Elementary Geometry of Differentiable Curves:An Undergraduate Introduction" book on page 6:

"Example 1.4: Recall that the component of a vector in the direction of a unit vector is the vector . (Example 1.1)

It is useful to express this in complex notation. Note that for any vectors we have : in particular when

b is a unit vector (i.e. ) we have ."

What I don't understand is why and how is: ?

The author said(given above) and I quote "for any vectors" this is true.

Can anyone kindly shed light on this and explain a bit why this is so?

Re: Can 2 times the dot product of 2 complex numbers equal to this?

Quote:

Originally Posted by

**x3bnm** "Example 1.4: Recall that the component of a vector

in the direction of a unit vector

is the vector

. (Example 1.1)

It is useful to express this in complex notation. Note that for any vectors

we have

: in particular when

b is a unit vector (i.e.

) we have

."

What I don't understand is why and how is:

?

Suppose that .

You can show that .

But as vectors

Re: Can 2 times the dot product of 2 complex numbers equal to this?

We have

The real part is just

Re: Can 2 times the dot product of 2 complex numbers equal to this?

Thanks a lot Plato and Vlasev. That makes sense now. Again Thanks.