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I have read a lot about why sign conventions are used in ray optics. It is because the formulas are different for different lens and object. My teacher also said that sign conventions are like the coordinate system. In the equations, we specify the coordinates of the object. lens and image. This is why we use sign convention again in the formula because the questions only specify the distance

But why is that when we apply sign convention, all the different formulas magically become one. Can someone give another example where taking a coordinate system resolves these difficulties ?

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That is because without sign convention even the direction, location have to become the part or component of the equation. So just for one lens, you might have to remember 4 or more formulas but when you take into consideration the sign convention then what you choose as your variables in the equation are not mere magnitudes but also magnitudes with direction.

For example take the mirror formula:

The one in textbook might say: $$1/f = 1/v + 1/u$$ as a general formula.

But lets say you didn't use convention then: the same formula for concave mirror with object on the same side as incident light: $1/f = 1/u +- 1/v$
(+/- depending on location of image which inturn depends on location of object)

for convex mirror: $-1/f = 1/u - 1/v$

for concave mirror but object within focus: $1/f = 1/u - 1/v$

for concave mirror with object below principal axis $1/f = -1/u +- 1/v$ and many more......

As for other examples asked by you:

Galileo's equations of motion and projectile motion equations depend on the convention of cartesian coordinates,

in equations involving charge and in electricity direction of electrons is considered negative and direction opposite is taken positive (which is the direction of current-again by convention) instead of having two equations- one for protons and one for electrons. So you get it - why sign conventions give generalized formulas!!! (Hopefully through a good example) See how they "magically" yet logically become one.

And pretty much for all equations in physics involving vectors, you have 2 options:

1) Create diverse formulas taking into account all the diverse directions possible - where variables of equations represent magnitude only- which means you have 100 times the formulas we know right now (don't believe the number "100 times" - just to give you the idea)

2) Create equations that have variables without direction being a part of the equation (basically meaning that variable say x also involves a direction) and then learn a simple convention that gives you the direction part and plug in the number with the direction given by convention into the equation.

The second method turns out a lot more easier and more practical. The second method and Sign conventions basically generalize formulas because they ignore the diversities produced due to multiple directions vectors can have.

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