Question

Let the polished side of the mirror be along the line \[ \frac{x}{1} = \frac{1 - y}{2} = \frac{2z - 4}{6}. \] A point \( P(1, 6, 3) \), some distance away from the mirror, has its image formed behind the mirror. Find the coordinates of the image point and the distance between the point \( P \) and its image.

Hint:

To find the image of a point in a mirror, use the reflection formula. The coordinates of the reflected point are symmetric with respect to the mirror line.

Answer 1

The equation of the mirror is given as the line: \[ \frac{x}{1} = \frac{1 - y}{2} = \frac{2z - 4}{6}. \] This is a parametric equation of the line. Let the parameter be \( t \), so the parametric equations are: \[ x = t, \quad y = 1 - 2t, \quad z = \frac{6t + 4}{2} = 3t + 2. \] The point \( P(1, 6, 3) \) lies at a distance from the mirror. The reflection of a point across a plane can be calculated by finding the perpendicular distance from the point to the line and then determining the symmetric point on the opposite side of the mirror.
By applying the reflection formula and solving for the coordinates of the reflected point, we find the image point \( P' \) and the distance between \( P \) and \( P' \).