vector components and dot product with unit vector

Question

$E_{0}\hat z=\vec E_{0}rcos\theta=E_{0}cos\theta\, \hat r$

and

$E_{0}cos\theta\, \hat r\circ \hat r =E_{0}cos\theta$

This just doesn't look right to me for some reason...

Answer 1

At the beginning you consider the projection of the vector $v=E_0z$ on the radial direction defined by $r$, denoting by $r$ the unit vector along the radial direction, by $z$ the unit vector along the $z$-direction and by $\theta$ the angle between the two directions. If the projection- which is a vector-is denoted by $p=E_0\cos \theta~ r$, then $$\langle E_0\cos \theta~ r, r \rangle = E_0\cos \theta \langle r, r \rangle = E_0\cos \theta$$

as $\langle r,r\rangle =\|r\|^2=1$ by definition.