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 mirror's equation is represented by the line: \[ \frac{x}{1} = \frac{1 - y}{2} = \frac{2z - 4}{6}. \] This can be expressed parametrically with parameter \( t \) as: \[ x = t, \quad y = 1 - 2t, \quad z = \frac{6t + 4}{2} = 3t + 2. \] The point \( P(1, 6, 3) \) is at a distance from this mirror. The reflection of a point across a plane involves calculating the perpendicular distance to the plane and then locating the symmetric point. By using the reflection formula and solving for the reflected point \( P' \), we can determine the distance between \( P \) and \( P' \).