linguistic:paraphrasing, information:extraction, information:highlighting
Reflection across a line in the plane

Reflection across an arbitrary line through the origin in two dimensions can be described by the following formula

where v {\displaystyle v} denotes the vector being reflected, l {\displaystyle l} denotes any vector in the line across which the reflection is performed, and v ⋅ l {\displaystyle v\cdot l} denotes the dot product of v {\displaystyle v} with l {\displaystyle l}. Note the formula above can also be written as

saying that a reflection of v {\displaystyle v} across l {\displaystyle l} is equal to 2 times the projection of v {\displaystyle v} on l {\displaystyle l}, minus the vector v {\displaystyle v}. Reflections in a line have the eigenvalues of 1, and −1.