Editing Real projective plane (section)

=== Points, lines, and planes ===
[[Image:Proj geom1.PNG|right]]
A line in '''P'''<sup>2</sup> can be represented by the equation ''ax'' + ''by'' + ''cz'' = 0. If we treat ''a'', ''b'', and ''c'' as the column vector '''ℓ''' and ''x'', ''y'', ''z'' as the column vector '''x''' then the equation above can be written in matrix form as:
: '''x'''<sup>T</sup>'''ℓ''' = 0 or '''ℓ'''<sup>T</sup>'''x''' = 0.

Using vector notation we may instead write '''x''' &sdot; '''ℓ''' = 0 or '''ℓ''' &sdot; '''x''' = 0.

The equation ''k''('''x'''<sup>T</sup>'''ℓ''') = 0 (which k is a non-zero scalar) sweeps out a plane that goes through zero in '''R'''<sup>3</sup> and ''k''(''x'') sweeps out a line, again going through zero. The plane and line are [[linear subspace]]s in [[real coordinate space|'''R'''<sup>3</sup>]], which always go through zero.
{{Clear}}