Editing Analytic geometry (section)

===Lines and planes===
{{main|Line (geometry)|Plane (geometry)}}
Lines in a [[Cartesian plane]], or more generally, in [[affine coordinates]], can be described algebraically by ''linear'' equations. In two dimensions, the equation for non-vertical lines is often given in the ''[[slope-intercept form]]'':
<math display="block"> y = mx + b </math>
where:
* ''m'' is the [[slope]] or [[slope|gradient]] of the line.
* ''b'' is the [[y-intercept]] of the line.
* ''x'' is the [[independent variable]] of the function ''y'' = ''f''(''x'').

In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it (the [[normal vector]]) to indicate its "inclination".

Specifically, let <math>\mathbf{r}_0</math> be the position vector of some point <math>P_0 = (x_0,y_0,z_0)</math>, and let <math>\mathbf{n} = (a,b,c)</math> be a nonzero vector. The plane determined by this point and vector consists of those points <math>P</math>, with position vector <math>\mathbf{r}</math>, such that the vector drawn from <math>P_0</math> to <math>P</math> is perpendicular to <math>\mathbf{n}</math>. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the desired plane can be described as the set of all points <math>\mathbf{r}</math> such that
<math display="block">\mathbf{n} \cdot (\mathbf{r}-\mathbf{r}_0) =0.</math>
(The dot here means a [[dot product]], not scalar multiplication.)
Expanded this becomes
<math display="block"> a (x-x_0)+ b(y-y_0)+ c(z-z_0)=0,</math>
{{cn span|text=which is the ''point-normal'' form of the equation of a plane. |date=April 2022}} This is just a [[linear equation]]:
<math display="block"> ax + by + cz + d = 0, \text{ where } d = -(ax_0 + by_0 + cz_0).</math>
Conversely, it is easily shown that if ''a'', ''b'', ''c'' and ''d'' are constants and ''a'', ''b'', and ''c'' are not all zero, then the graph of the equation
<math display="block"> ax + by + cz + d = 0,</math>
{{cn span|text=is a plane having the vector <math>\mathbf{n} = (a,b,c)</math> as a normal.|date=April 2022}} This familiar equation for a plane is called the ''general form'' of the equation of the plane.<ref>{{citation
 | last1 = Vujičić | first1 = Milan
 | last2 = Sanderson | first2 = Jeffrey
 | editor-first1 = Milan
 | editor-first2 = Jeffrey
 | editor-last1 = Vujičić
 | editor-last2 = Sanderson
 | doi = 10.1007/978-3-540-74639-3
 | page = 27
 | publisher = Springer
 | title = Linear Algebra Thoroughly Explained
 | year = 2008| isbn = 978-3-540-74637-9
 }}</ref>

In three dimensions, lines can ''not'' be described by a single linear equation, so they are frequently described by [[parametric equation]]s:
<math display="block"> x = x_0 + at </math>
<math display="block"> y = y_0 + bt </math>
<math display="block"> z = z_0 + ct </math>
where:
* ''x'', ''y'', and ''z'' are all functions of the independent variable ''t'' which ranges over the real numbers.
* (''x''<sub>0</sub>, ''y''<sub>0</sub>, ''z''<sub>0</sub>) is any point on the line.
* ''a'', ''b'', and ''c'' are related to the slope of the line, such that the [[vector (geometric)|vector]] (''a'', ''b'', ''c'') is parallel to the line.