Plane Mirrors — Question 1

Question 1

A translated plane mirror reflects only the normal coordinate

A plane mirror occupies x=3 cm. An object point is at (1,4,2) cm and an observer is at (0,0,2) cm. Use fixed Cartesian coordinates.

1. Find the virtual image coordinates.

2. Find the straight-line apparent distance from observer to image and compare object-image separation with object-mirror distance.

Question 1 - Solution

A translated plane mirror reflects only the normal coordinate

(a) Method and calculation

$$x_I=2x_M-x_O=6-1=5,\quad y_I=y_O,\quad z_I=z_O.$$$$\boxed{\mathbf r_I=(5,4,2)\,\mathrm{cm}}.$$

(b) Interpretation and further calculation

$$\boxed{d_{EI}=\sqrt{(5-0)^2+(4-0)^2+(2-2)^2}=\sqrt{41}\,\mathrm{cm}}.$$$$\boxed{d_{OI}=4\,\mathrm{cm}=2d_{OM}}.$$

Physical check

The coordinates parallel to the mirror are unchanged; the apparent distance to an off-axis observer is not simply twice the normal object distance.