In Euclidean geometry, a reflection is a linear operation σ on
with σ2 = E, the identity map. This property of σ is called involution. An involutory operator is non-singular and σ−1 = σ. Reflecting twice an arbitrary vector brings back the original vector :

The operation σ is an isometry of
onto itself, which means that it preserves inner products and that its inverse is equal to its adjoint,

Hence reflection is also symmetric: σT = σ. From (det(σ))2 = det(E) = 1 follows that isometries have determinant ±1. Those with positive determinant are rotations, while reflections have determinant −1. Because σ is symmetric it has real eigenvalues; since the extension of an isometry to a complex space is unitary, its (complex) eigenvalues have modulus 1. It follows that the eigenvalues of σ are ±1. The product of the eigenvalues being its determinant, −1, the sets of eigenvalues of σ are either {1, 1, −1}, or {−1, −1, −1}. An operator with the latter set of eigenvalues is equal to −E, minus the identity operator. This operator is known alternatively as inversion, reflection in a point, or parity operator. An operator with the former set of eigenvalues is reflection in a plane. Reflections in a plane are the subject of this article.
Sometimes one finds the concept of "reflections in a line", these are rotations over 180°, see rotation matrix.
PD Image Fig. 1. The vector

goes to

under reflection in a plane. The unit vector

is normal to mirror plane.
Reflection in a plane
If
is a unit vector normal (perpendicular) to a plane—the mirror plane—then
is the projection of
on this unit vector. From the figure it is evident that

If a non-unit normal
is used then substitution of

gives the mirror image,

Sometimes it is convenient to write this as a matrix equation. Introducing the dyadic product, we obtain
![{\displaystyle {\vec {\mathbf {r} }}\,'=\left[\mathbf {E} -{\frac {2}{n^{2}}}{\vec {\mathbf {n} }}\otimes {\vec {\mathbf {n} }}\right]\;{\vec {\mathbf {r} }},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb112d732fbaf7c81a05f819922f6b4d713dc646)
where E is the 3×3 identity matrix.
Dyadic products satisfy the matrix multiplication rule
![{\displaystyle [{\vec {\mathbf {a} }}\otimes {\vec {\mathbf {b} }}]\,[{\vec {\mathbf {c} }}\otimes {\vec {\mathbf {d} }}]=({\vec {\mathbf {b} }}\cdot {\vec {\mathbf {c} }}){\big (}{\vec {\mathbf {a} }}\otimes {\vec {\mathbf {d} }}{\big )}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/037a6b38897a0d8a8d7787a0ea60a4e18d091281)
By the use of this rule it is easily shown that
![{\displaystyle \left[\mathbf {E} -{\frac {2}{n^{2}}}{\vec {\mathbf {n} }}\otimes {\vec {\mathbf {n} }}\right]^{2}=\mathbf {E} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd65ff7a8f5970ebba0e77e1a5680c6192a28266)
which confirms that reflection is involutory.
PD Image Fig. 2. The vector

goes to

under reflection
Reflection in a plane not through the origin
In Figure 2 a plane, not containing the origin O, is considered that is orthogonal to the vector
. The length of this vector is the distance from O to the plane.
From Figure 2, we find

Use of the equation derived earlier gives

And hence the equation for the reflected pair of vectors is,

where
is a unit vector normal to the plane. Obviously
and
are proportional, they differ only by scaling. Therefore, the equation can be written solely in terms of
,

Two consecutive reflections
PD Image Fig. 3. Two reflections. Left drawing: 3-dimensional drawing. Right drawing: view along the PQ axis, drawing projected on the plane through ABC. This plane intersect the line PQ in the point P′
Two consecutive reflections in two intersecting planes give a rotation around the line of intersection. This is shown in Figure 3, where PQ is the line of intersection.
The drawing on the left shows that reflection of point A in the plane through PMQ brings the point A to B. A consecutive reflection in the plane through PNQ brings B to the final position C. In the right-hand drawing it is shown that the rotation angle φ is equal to twice the angle between the mirror planes. Indeed, the angle ∠ AP'M = ∠ MP'B = α and ∠ BP'N = ∠ NP'C = β. The rotation angle ∠ AP'C ≡ φ = 2α + 2β and the angle between the planes is α+β = φ/2.
It is obvious that the product of two reflections is a rotation. Indeed, a reflection is an isometry and has determinant −1. The product of two isometric operators is again an isometry and the rule for determinants is det(AB) = det(A)det(B), so that the product of two reflections is an isometry with unit determinant, i.e., a rotation.
Let the normal of the first plane be
and of the second
, then the rotation is represented by the matrix
![{\displaystyle \left[\mathbf {E} -{\frac {2}{t^{2}}}{\vec {\mathbf {t} }}\otimes {\vec {\mathbf {t} }}\right]\,\left[\mathbf {E} -{\frac {2}{s^{2}}}{\vec {\mathbf {s} }}\otimes {\vec {\mathbf {s} }}\right]=\mathbf {E} -{\frac {2}{t^{2}}}{\vec {\mathbf {t} }}\otimes {\vec {\mathbf {t} }}-{\frac {2}{s^{2}}}{\vec {\mathbf {s} }}\otimes {\vec {\mathbf {s} }}+{\frac {4}{t^{2}s^{2}}}({\vec {\mathbf {t} }}\cdot {\vec {\mathbf {s} }})\;{\big (}{\vec {\mathbf {t} }}\otimes {\vec {\mathbf {s} }}{\big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/454d07547f582382357684b9dc6463f3ff0ecbbc)
The (i,j) element if this matrix is equal to

This formula is used in vector rotation.