On my website, users can vote on whether they like or dislike various things, and then they can write either positive or negative opinions. Historically, has positivity been associated with the right-hand or left-hand side of the UI?
There are several things to consider here.
Do you want to fill your website with negative votes ?
Most of the pages and languages are read from left to right. Just like placing an OK button on the left of a cancel button to get more attention , a positive vote can be placed on the left of a negative vote.
Examples and differences.
Is your website function dependent on the positive votes?
Say you have a feed that populates items based in positive votes alone , in that case you should give more attention to positive vote option ( may be with bold text , a different colour or just breaking the pattern )
Is it a place where both positive and negative votes are important?
In that case you can place positive vote to the left of negative vote with equal importance (may be with a colour change to depict the difference in purpose ).
not helpful votes in a review website.
Smartphone context would make the matter easier because you would think distance to the thumb.
For desktop sized web, key would be making the two voting options dramatically different visually. Size, color, shape, iconography, font all need to communicate the difference between positive and negative answers.
For a grand example, compare Windows and Mac systems.
Windows generally place positive buttons on the left and negative, right. And Mac systems (including iOS) does it the opposite. In fact, Android dialogs also tend to have the positive or confirm buttons on the right.
Look at whatever data you have and see if your primary audience is using either of the said systems. What is intuitive is very subjective as it's born out of habit. So, capitalize on whatever habit your primary audience has acquired in using other systems.
I think that there is a problem with this question:
Indeed, a positive vote is the opposite of a negative vote. Thus, If I see two buttons (or links), one on the left and one one the right, this suggests that I can clic on both, so I can vote positively and negatively for the same stuff.
If you want two buttons, they must be more than opposites and some different rewards should appear for each choice.
In your simple case, I suggest you to only show a positive button (eg. a like).