Q.State the New Cartesian sign convention used in ray optics for spherical mirrors and lenses, and explain why such a convention is needed at all.

A single algebraic formula such as the mirror formula is meant to work for a concave mirror and a convex mirror, for a real object and (later) a virtual object, and for a real or a virtual image, all without the student having to remember a different formula or a different rule for each case. The New Cartesian Sign Convention makes this possible by fixing, once and for all, how every distance and height in a ray-optics problem is to be signed: the pole of the mirror (or optical centre of the lens) is taken as the origin; the principal axis is taken as the x-axis; the object is always drawn to the left, so incident light is taken to travel left-to-right, and distances measured in this same, incident-light direction are positive while distances measured against it (to the left) are negative; heights measured upward from the axis are positive and downward are negative. Once this is fixed, a real object (always to the left) always has $u<0$; a concave mirror's focus (in front of the mirror, same side as the object) has $f<0$, while a convex mirror's focus (behind the mirror) has $f>0$; and so on. Substituting the correctly-signed numbers into one formula then automatically produces the correct sign, and hence the correct physical meaning, for every unknown.

State the reference point (pole/optical centre), the positive direction (direction of incident light) and the rule for heights, in that order.

• Forgetting that the object distance $u$ is always negative for a real object, regardless of mirror type.
• Mixing up which mirror (concave/convex) gets a negative vs. positive focal length.
• Applying the convention inconsistently between the object side and the image side of the same problem.