Editing Real projective plane (section)

==== Lines joining points and intersection of lines (using duality) ====
The equation {{nowrap|'''x'''<sup>T</sup>'''ℓ''' {{=}} 0}} calculates the [[dot product|inner product]] of two column vectors. The inner product of two vectors is zero if the vectors are [[orthogonal]]. In  '''P'''<sup>2</sup>, the line between the points '''x'''<sub>1</sub> and '''x'''<sub>2</sub> may be represented as a column vector '''ℓ''' that satisfies the equations {{nowrap|'''x'''<sub>1</sub><sup>T</sup>'''ℓ''' {{=}} 0}} and {{nowrap|'''x'''<sub>2</sub><sup>T</sup>'''ℓ''' {{=}} 0}}, or in other words a column vector '''ℓ''' that is orthogonal to '''x'''<sub>1</sub> and '''x'''<sub>2</sub>. The [[cross product]] will find such a vector: the line joining two points has homogeneous coordinates given by the equation {{nowrap|'''x'''<sub>1</sub> × '''x'''<sub>2</sub>}}. The intersection of two lines may be found in the same way, using duality, as the cross product of the vectors representing the lines, {{nowrap|'''ℓ'''<sub>1</sub> × '''ℓ'''<sub>2</sub>}}.