Reflections using vectors (HL)

Reflecting point $P$ across a line or a plane, and some extensions that are probably beyond the syllabus. It’s more important to visualize the steps and draw a diagram, rather than memorizing them.

To find the reflection point $R$, often we need to find the nearest point $Q$ on the line or plane.

Then utilize the midpoint formula.

Last edited: 2026-06-09 added diagrams and made the solutions more consistent

Contents

• Reflect point across a point
• Reflect point across a plane
• Reflect a line across a parallel plane
• Reflect a line across a non-parallel plane
• Reflect point across across a line

Reflect point across a point

The reflection of point $P$ across point $Q$, is

meaning that $P$ and its reflection average to $Q$, or that you start from the point then add twice of the displacement vector to $Q$.

Similarly to reflect a line or a plane across a point, reflect a point on the line or plane. The new line or plane has the same direction vector or normal vector as before, but passing through the reflected point.

Reflect point across a plane

Given some plane $ax + by + cz = d$, the line through $P$ perpendicular to the plane is

Solve for the $\lambda_0$ at the point of intersection.

$P$ is at $\lambda = 0$, the intersection $Q$ is at $\lambda = \lambda_0$, and the reflected point is at $\lambda = 2\lambda_0$.

Reflect a line across a parallel plane

Reflect the given point on the line, across the plane. The reflected line has the same direction vector as before but goes through the reflected point.

Similar ideas exist for reflecting a line across a parallel line, a plane across a parallel line, and a plane across a parallel plane.

Reflect a line across a non-parallel plane

Find the intersection.

Reflect the given point on the line, across the plane. The reflected line goes through the intersection and the reflected point.

Alternatively you can also reflect any two points on the line and find the line through those new points.

Reflect point across across a line

Finding the reflection across $\bm r = \overrightarrow{OA} + \lambda \bm d$, given $\overrightarrow{PA} = \overrightarrow{OA} - \overrightarrow{OP}$.

First find $\lambda_0$ such that $\left(\overrightarrow{PA} + \lambda_0 \bm d\right) \cdot \bm d = 0$, where $\lambda_0 \bm d$ is a vector from $P$ to the nearest point $Q$, and the vector sum is perpendicular to the line.

The point on the line nearest to $\overrightarrow{OP}$ is

As the point and its reflection average to $\overrightarrow{OR}$, the reflected point is

You can also reflect lines and planes across a (non-parallel) line, but they get tedious. One way is finding the equations of the line (or plane) from the reflections of two (or three) points.