Editing Analytic geometry (section)

==Distance and angle==
{{main|Distance|Angle}}
[[File:Distance Formula.svg|thumb|right|250px|The distance formula on the plane follows from the Pythagorean theorem.]]

In analytic geometry, geometric notions such as [[distance]] and [[angle]] measure are defined using [[formula]]s.  These definitions are designed to be consistent with the underlying [[Euclidean geometry]].  For example, using [[Cartesian coordinates]] on the plane, the distance between two points (''x''<sub>1</sub>,&nbsp;''y''<sub>1</sub>) and (''x''<sub>2</sub>,&nbsp;''y''<sub>2</sub>) is defined by the formula
<math display="block">d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2},</math>
which can be viewed as a version of the [[Pythagorean theorem]].  Similarly, the angle that a line makes with the horizontal can be defined by the formula
<math display="block">\theta = \arctan(m),</math>
where ''m'' is the [[slope]] of the line.

In three dimensions, distance is given by the generalization of the Pythagorean theorem:
<math display="block">d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2+ (z_2 - z_1)^2},</math>
while the angle between two vectors is given by the [[dot product]]. The dot product of two Euclidean vectors '''A''' and '''B''' is defined by<ref name="Spiegel2009">{{cite book |author1=M.R. Spiegel |author2=S. Lipschutz |author3=D. Spellman |title= Vector Analysis (Schaum's Outlines)|edition= 2nd |year= 2009|publisher= McGraw Hill|isbn=978-0-07-161545-7}}</ref>
<math display="block">\mathbf A\cdot\mathbf B \stackrel{\mathrm{def}}{=} \left\|\mathbf A\right\| \left\|\mathbf B\right\| \cos\theta,</math>
where ''θ'' is the [[angle]] between '''A''' and '''B'''.