Editing Reflection (mathematics) (section)

==Reflection across a line in the plane==
{{Further|topic=reflection of light rays|Specular reflection#Direction of reflection}}
{{see also|180-degree rotation}}

Reflection across an arbitrary line through the origin in [[two dimensions]] can be described by the following formula
:<math>\operatorname{Ref}_l(v) = 2\frac{v \cdot l}{l \cdot l}l - v,</math>

where <math>v</math> denotes the vector being reflected, <math>l</math> denotes any vector in the line across which the reflection is performed, and <math>v\cdot l</math> denotes the [[dot product]] of <math>v</math> with <math>l</math>. Note the formula above can also be written as
:<math>\operatorname{Ref}_l(v) = 2\operatorname{Proj}_l(v) - v,</math>
saying that a reflection of <math>v</math> across <math>l</math> is equal to 2 times the [[vector projection|projection]] of <math>v</math> on <math>l</math>, minus the vector <math>v</math>.  Reflections in a line have the eigenvalues of 1, and −1.