- #1

- 479

- 20

Suppose that ##\phi## and ##\theta## are defined as follows:

##\phi## is the angle between the position vector of a point and the ##z##-axis. ##\theta## is the angle between the projection of that vector onto the ##xy##-plane and the ##x##-axis.

Why exactly do we write ##x = r \sin{\phi} \cos{\theta}##?

I know that we first project the position vector onto the ##xy##-plane before projecting that projection onto the ##x##-axis. But if that were the case, shouldn't we write ##x = |r \sin{\phi}| \cos{\theta}##?

I'm confused because the quantity ##r \sin{\phi}## is, on occasion, negative. I thought we're only allowed to project "lengths" or "positive scalars". I am fully aware of the fact that the projection can itself be negative, but the whole notion that the length (not the final projection) which is to be projected can be negative isn't so intuitive.

Take for example a circle of radius ##r## centred at the origin. I'll define ##t## as the angle made with the ##x##-axis. ##x = r \cos{\theta}##. ##x## is allowed to be positive or negative, but ##r## is always positive.