Is the carbon atom in the carbon dioxide molecule partially positive? I had a question about nonpolar molecules that have symmetrical dipole vectors.

Let's take $\ce{CO2}$ as an example. Each of the $\ce{C=O}$ bonds are pulling in the opposite way. My teacher says that this causes all atoms in $\ce{CO2}$ to be equally charged as there mustn't be any net 'force'.

However, I disagree. Intuitively, it seems that the oxygen atoms would pull electron density away from the central carbon atom and make the carbon atom slightly positive and the oxygen atoms slightly negative, like so:

$$\large\ce{\overset{\small\delta-}{O}=\overset{\small\delta+}{C}=\overset{\small\delta-}{O}}$$

This process should make the carbon atom slightly positive and the oxygen atoms slightly negative. However, if I was right, then why don't we say $\ce{CO2}$ has a dipole (there is a separation of charge)? Perhaps I may have the wrong definition of a dipole.
You're correct in assuming that the carbon atom in $\ce{CO2}$ has a partial positive charge. This is because the oxygen atoms are much more electronegative, so they pull the electrons away from the carbon atom. However, this molecule is still nonpolar. This is because, when you draw a dipole moment, you have to take all bonds into account. Take water for example:



In this molecule, there are two bonds, each with their own dipole. But these cancel out like any other vectors, leaving you with a vertical net dipole. The dipoles in carbon dioxide cancel out in a similar way; however, they cancel each other completely, because the bond is linear, not bent like in water:



This produces a zero net dipole, rendering the molecule nonpolar.