Difference between revisions of "Perpendicular"

 
(4 intermediate revisions by 2 users not shown)
Line 1: Line 1:
{{stub}}
+
Being '''perpendicular''' is a property of lines in a plane. Generally, when the term is used, it refers to the definition of perpendicular in [[Euclidean]] geometry.
 
 
Two [[line]]s <math>l</math> and <math>m</math> are said to be '''perpendicular''' if they intersect in [[right angle]]s. We denote this relationship by <math>l \perp m</math>.  In the [[Cartesian coordinate system]], a line with [[slope]] <math>m</math> is perpendicular to every line with slope <math>-\frac{1}{m}</math> and no others.
 
  
 +
==Definition==
 +
Two [[line]]s <math>l</math> and <math>m</math> are said to be '''perpendicular''' if they intersect in [[right angle]]s.  We denote this relationship by <math>l \perp m</math>. 
  
 +
===For non-linear objects===
 
One can also discuss perpendicularity of other objects.  If a line <math>l</math> intersects a plane <math>P</math> at a point <math>A</math>, we say that <math>l \perp P</math> if and only if for ''every'' line <math>m</math> in <math>P</math> passing through <math>A</math>, <math>l \perp m</math>.   
 
One can also discuss perpendicularity of other objects.  If a line <math>l</math> intersects a plane <math>P</math> at a point <math>A</math>, we say that <math>l \perp P</math> if and only if for ''every'' line <math>m</math> in <math>P</math> passing through <math>A</math>, <math>l \perp m</math>.   
  
 
If a plane <math>P</math> intersects another plane <math>Q</math> in a line <math>k</math>, we say that <math>P \perp Q</math> if and only if:
 
If a plane <math>P</math> intersects another plane <math>Q</math> in a line <math>k</math>, we say that <math>P \perp Q</math> if and only if:
 
for line <math>l \in P</math> and <math>m \in Q</math> passing through <math>A \in k</math>, <math>l \perp k</math> and <math>m \perp k</math> implies <math>l \perp m</math>.
 
for line <math>l \in P</math> and <math>m \in Q</math> passing through <math>A \in k</math>, <math>l \perp k</math> and <math>m \perp k</math> implies <math>l \perp m</math>.
 +
 +
==Coordinate Plane==
 +
Two linear graphs in the Cartesian coordinate plane are perpendicular if and only if one's [[slope]] is the negative reciprocal of the other's. This means that their slopes must have a product of <math>-1</math>.
 +
 +
==See Also==
 +
*[[Parallel]]
 +
*[[Skew]]
 +
[[Category:Definition]]
 +
[[Category:Geometry]]

Latest revision as of 14:33, 20 October 2007

Being perpendicular is a property of lines in a plane. Generally, when the term is used, it refers to the definition of perpendicular in Euclidean geometry.

Definition

Two lines $l$ and $m$ are said to be perpendicular if they intersect in right angles. We denote this relationship by $l \perp m$.

For non-linear objects

One can also discuss perpendicularity of other objects. If a line $l$ intersects a plane $P$ at a point $A$, we say that $l \perp P$ if and only if for every line $m$ in $P$ passing through $A$, $l \perp m$.

If a plane $P$ intersects another plane $Q$ in a line $k$, we say that $P \perp Q$ if and only if: for line $l \in P$ and $m \in Q$ passing through $A \in k$, $l \perp k$ and $m \perp k$ implies $l \perp m$.

Coordinate Plane

Two linear graphs in the Cartesian coordinate plane are perpendicular if and only if one's slope is the negative reciprocal of the other's. This means that their slopes must have a product of $-1$.

See Also

Invalid username
Login to AoPS