I Is the Dot Product of Unit Vectors Related to Magnitudes and Angle Between Them?

[Abdulwahab Hajar]

Okay so I understand that in order to represent a vector which is in cartesian coordinates in spherical coordinates.. we use the transformation which is obtained by dotting the unit vectors.
So my question goes like this:
when we dot for example the unit vector ar^ with x^ we obtain sin(theta) * cos(phi), however can't the dot product be interpreted as the magnitudes multiplied by the cos of the angle between them.
In this case the magnitudes are 1 because they are unit vectors but how can sin(theta) * cos(phi) equal cos(angle between ar^ and x^)
I know my notation sucks please pardon me it's my first time posting... I have no notation at all :(

Thank you for the help

 

[Simon Bridge]

If you identify the angle between $\vec r = r\hat r$ and $\vec x = x\hat\imath$ (you OK with i-j-k unit vectors?) for an arbitrary $\vec r$ as $\alpha$ to distinguish it from the $\theta$ and $\phi$ of the spherical polar coordinates... then $\hat r\cdot \hat\imath = \cos\alpha$ right?

You can express $\cos\alpha$ in terms of $\theta$ and $\phi$.
Give it a go. ie. try first for $\theta=\pi/2$ and $\phi >0$, then for $\phi=0$ and $0<\theta<\pi/2$ ... then combine the results.

 

[Abdulwahab Hajar]

Awesome haha I actually got it :)
thanks a million sir