What is the correct sign convention for the lens and mirror formula?

[rishch]

So we are learning the lens formula and I have two textbooks, my school textbook and another one that is much more detailed. My school textbook gives the lens formula as:-

1/f=1/v-1/u

while the other one gives

1/f=1/v+1/u (v is image distance and u is object distance)

There both different! I looked them some more and the problem is that my school textbook follows some "New Cartesian" sign convention. Imagine the optical center as the origin and the principal axis as the x-axis and the vertical line down the mirror as the y axis. So left is negative, right is positive, up is positive and down is negative. Just like a graph. The other one does the same except that even if the object is on the left they take it as positive.

I went to Khan Academy and he had a proof and he came up with the second one, different from the one in my school textbook. I tried doing the proofs on my own but I don't know how to include negative distances in geometry. Do you take the lengths as negative? Then you can't use similarity of triangles because some sides have negative length and others, positive. How do you work with negative distances? Getting really confused :/

 

[Simon Bridge]

The differences are due to different sign conventions all right.
I've never been able to keep them straight and I always rely on a quick sketch of the situation as a reality check.

Note: Triangles are similar if all the angles are the same ... with negative distances, you get a negative scale factor.

 

[Simon Bridge]

The differences are due to different sign conventions all right.
I've never been able to keep them straight and I always rely on a quick sketch of the situation as a reality check.

Note: Triangles are similar if all the angles are the same ... with negative distances, you get a negative scale factor.

 

[rishch]

Actually I find the cartesian plane sign convention really good as it's easy to remember if you just thing of it like a graph or number line. Left is negative and right is positive. Yes but during proofs one pair of corresponding sides may have a negative scale factor and one may have a positive scale factor. I managed to come up with an alternative neat solution to the problem after a bit of thinking though.