Reflection of a line in a plane

Reflection of a Line in a Plane

Definition

The reflection of a line in a plane refers to the process of creating a mirror image of the line across a plane.
The plane can be considered as a ‘mirror’ and the reflected line is the image you would see if the original line was ‘reflecting’ off this mirror.
After the reflection, every point on the original line and its reflected image are equidistant from the plane of reflection.

Method

To reflect a line in a plane, begin by finding a normal vector to the plane. This vector is perpendicular to the plane.
You calculate the position vector of any point on the line reflected in the plane by subtracting twice the line’s projection onto the normal vector from the position vector of the point on the original line.
This method effectively moves the point on the original line to its reflected position across the plane.

Applications

Reflections are essential in building the foundations for understanding symmetry in shapes and space, used widely in geometry and algebra.
These concepts have further applications in areas like physics, computer graphics, and engineering designs, where understanding reflections can aid in problem-solving.

Example

For instance, if a line L with equation r = a + tb (where r, a and b are vectors, t is a scalar) is reflected in the plane with normal vector n, a point P on the reflected line L’ has position vector r’ given by: r’ = r - 2((r - a).n /

n

^2) * n

This formula calculates the position vector of the point on the reflected line, which you can use to find the equation of the reflected line.

Key Points

The reflection of a line in a plane uses the plane as a ‘mirror’ to create a symmetrical image of the original line.
Finding the reflection involves the application of vectors, specifically concepts of position vectors and normal vectors.
Reflections are a fundamental concept in both geometry and algebra, carry practical relevance in physical and digital applications.