Reflections Across Lines in R^2

Reflection in Coordinates Adapted to a Line

A reflection across a line through the origin keeps points on the line fixed and reverses the perpendicular component of every vector. This description is simple when coordinates are measured parallel and perpendicular to the line. The main task is to translate that action back into the usual coordinates in \(\mathbb{R}^2\). We will derive the matrix and then verify directly that it preserves lengths and reverses orientation.

Let \(\theta\) be the angle measured counterclockwise from the positive \(x\)-axis to the line. A direction vector along the line and a perpendicular unit vector are

Both vectors have length \(1\), and their dot product is \(-\cos\theta\sin\theta+\sin\theta\cos\theta=0\). Thus \((\mathbf{u},\mathbf{n})\) is an orthonormal basis of \(\mathbb{R}^2\). Every vector \(\mathbf{x}\) has a unique expression \(\mathbf{x}=a\mathbf{u}+b\mathbf{n}\), where \(a=\mathbf{u}^{\top}\mathbf{x}\) and \(b=\mathbf{n}^{\top}\mathbf{x}\). Reflection across the line keeps the \(a\)-component and changes the sign of the \(b\)-component.

To convert from the adapted basis to standard coordinates, let \(\mathbf{Q}\) have columns \(\mathbf{u}\) and \(\mathbf{n}\). Then \(\mathbf{Q}\) is orthogonal: its columns are orthonormal, so \(\mathbf{Q}^{\top}\mathbf{Q}=\mathbf{I}_2\). The coordinates of \(\mathbf{x}\) in the adapted basis are \(\mathbf{Q}^{\top}\mathbf{x}\). In those coordinates reflection is represented by \(\begin{bmatrix}1&0\\0&-1\end{bmatrix}\). Converting back gives the standard-coordinate matrix

Multiplying these matrices gives an explicit formula. Write \(c=\cos\theta\) and \(s=\sin\theta\), so \(\mathbf{Q}=\begin{bmatrix}c&-s\\s&c\end{bmatrix}\). First,

Therefore

The last equality uses the double-angle identities \(\cos(2\theta)=\cos^2\theta-\sin^2\theta\) and \(\sin(2\theta)=2\sin\theta\cos\theta\). The angle doubles because the orientation of the line affects both the direction of the component to preserve and the perpendicular direction to reverse.

Properties of the Reflection Matrix

Proof. The coordinate construction defines a linear map: in the basis \((\mathbf{u},\mathbf{n})\), it sends coordinates \((a,b)^{\top}\) to \((a,-b)^{\top}\). Thus it fixes every multiple of \(\mathbf{u}\) and reverses every multiple of \(\mathbf{n}\), which is precisely reflection across the line. The matrix formula follows from converting coordinates using \(\mathbf{Q}\), as shown above. Since \(\mathbf{Q}^{\top}\mathbf{Q}=\mathbf{I}_2\) and \(\mathbf{D}=\begin{bmatrix}1&0\\0&-1\end{bmatrix}\) satisfies \(\mathbf{D}^{\top}\mathbf{D}=\mathbf{I}_2\), we have \(\mathbf{S}_{\theta}^{\top}\mathbf{S}_{\theta}=\mathbf{Q}\mathbf{D}^{\top}\mathbf{Q}^{\top}\mathbf{Q}\mathbf{D}\mathbf{Q}^{\top}=\mathbf{Q}\mathbf{I}_2\mathbf{Q}^{\top}=\mathbf{I}_2\). So \(\mathbf{S}_{\theta}\) is orthogonal. Also \(\det\mathbf{Q}=c^2+s^2=1\), \(\det\mathbf{D}=-1\), and \(\det\mathbf{Q}^{\top}=1\), giving \(\det\mathbf{S}_{\theta}=-1\). Finally, the defining action gives \(\mathbf{S}_{\theta}\mathbf{u}=\mathbf{u}\) and \(\mathbf{S}_{\theta}\mathbf{n}=-\mathbf{n}\). \(\square\)

These properties provide useful checks. Orthogonality implies preservation of dot products and lengths, while determinant \(-1\) distinguishes a reflection from a rotation, whose determinant is \(1\). A further consequence is that applying the reflection twice returns the original vector: both coordinates \((a,b)\) undergo \((a,b)\mapsto(a,-b)\mapsto(a,b)\).

Proof. Write an arbitrary vector as \(\mathbf{x}=a\mathbf{u}+b\mathbf{n}\). By the definition of reflection, \(\mathbf{S}_{\theta}\mathbf{x}=a\mathbf{u}-b\mathbf{n}\). Equality with \(\mathbf{x}\) is equivalent to \(a\mathbf{u}-b\mathbf{n}=a\mathbf{u}+b\mathbf{n}\), hence \(2b\mathbf{n}=\mathbf{0}\). Since \(\mathbf{n}\) is a unit vector, it is nonzero, so this equation holds exactly when \(b=0\). In that case \(\mathbf{x}=a\mathbf{u}\) is on the line. Conversely, if \(\mathbf{x}=a\mathbf{u}\), the defining rule gives \(\mathbf{S}_{\theta}\mathbf{x}=a\mathbf{u}=\mathbf{x}\). \(\square\)

Computing Reflections

Angles, Checks, and a Common Sign Error

A line has no preferred direction: the angles \(\theta\) and \(\theta+\pi\) describe the same line. The formula respects this fact because \(\cos(2(\theta+\pi))=\cos(2\theta+2\pi)=\cos(2\theta)\) and \(\sin(2(\theta+\pi))=\sin(2\theta+2\pi)=\sin(2\theta)\). So the reflection matrix does not depend on which of the two opposite directions along the line is chosen.

When checking a computed reflection, two tests are especially effective. First, apply the same matrix a second time: the result must be the input, because the perpendicular component is reversed twice. Second, check that the norm is preserved; orthogonality guarantees \(\|\mathbf{S}_{\theta}\mathbf{x}\|=\|\mathbf{x}\|\). These checks detect many arithmetic errors, though neither replaces writing down the matrix and multiplication carefully.

Common mistake: using the rotation matrix instead of the reflection matrix. At \(\theta=\pi/4\), a sign error in the off-diagonal entries might produce \(\begin{bmatrix}0&-1\\1&0\end{bmatrix}\). This is a rotation matrix, not a reflection matrix: it sends \((1,0)^{\top}\) to \((0,1)^{\top}\), whereas reflection across \(y=x\) should send \((1,0)^{\top}\) to \((0,1)^{\top}\)—this single vector does not expose the error. Test instead \((0,1)^{\top}\): reflection should send it to \((1,0)^{\top}\), but the incorrect matrix sends it to \((-1,0)^{\top}\). The correct formula has equal off-diagonal entries, \(\sin(2\theta)\), because reflecting across \(y=x\) exchanges the two coordinates.

There is a useful algebraic way to distinguish the two kinds of transformation. For a reflection, \(\mathbf{S}_{\theta}^{2}=\mathbf{I}_2\) and \(\det\mathbf{S}_{\theta}=-1\). A nontrivial rotation generally does not square to the identity and has determinant \(1\). Thus the action on a particular vector can be checked against the geometric definition, while the square and determinant help identify the type of matrix.