Maximum power transfer theorem: Difference between revisions

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[[Image:Perpendicular-coloured.svg|right|thumb|250px|The segment AB is perpendicular to the segment CD because the two angles it creates (indicated in orange and blue) are each 90 degrees.]]
{{General geometry}}
 
{{Other uses}}
In elementary [[geometry]], the word '''perpendicular''' describes the relationship between two lines which meet at a [[right angle]].
 
A [[line (geometry)|line]] is said to be '''perpendicular''' to another line if the two lines intersect at a [[right angle]]. Explicitly, a first line is perpendicular to a second line if 1) the two lines meet and 2) at the point of intersection the [[straight angle]] on one side of the first line is cut by the second line into two [[congruence (geometry)|congruent]] [[angle]]s. Perpendicularity can be shown to be [[symmetric]], meaning if a first line is perpendicular to a second line, then the second line is also perpendicular to the first. For this reason, we may speak of two lines as being perpendicular (to each other) without specifying an order.  
 
Perpendicularity easily extends to segments and rays. For example, we say a line segment <math>\overline{AB}</math> is perpendicular to a line segment <math>\overline{CD}</math> if, when each is extended in both directions to form an infinite line, these two resulting lines are perpendicular in the sense above. In symbols, we write <math>\overline{AB} \perp \overline{CD}</math> to mean line segment AB is perpendicular to line segment CD.
 
A line is said to be '''perpendicular''' to a plane if it is perpendicular to every line in the plane that it intersects. Note that this definition depends on the definition of perpendicularity between lines.  
 
Two planes in space are said to be '''perpendicular''' if the [[dihedral angle]] at which they meet is a right angle (90 degrees).  
 
In fact, perpendicularity is one particular instance of the more general mathematical concept of [[orthogonality]]; perpendicularity is the orthogonality of classical geometric objects. Thus in advanced mathematics, the word perpendicular is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its [[Normal (geometry)|normal]].
 
==Construction of the perpendicular==
 
[[Image:Perpendicular-construction.svg|right|thumb|Construction of the perpendicular (blue) to the line AB through the point P.]]
 
To make the perpendicular to the line AB through the point P using [[compass and straightedge]], proceed as follows (see figure):
 
* Step 1 (red): construct a [[circle]] with center at P to create points A' and B' on the line AB, which are [[equidistant]] from P.
* Step 2 (green): construct circles centered at A' and B' having equal radius. Let Q and R be the points of intersection of these two circles.
* Step 3 (blue): connect Q and R to construct the desired perpendicular PQ.
To prove that the PQ is perpendicular to AB, use the [[Congruence (geometry)#Congruence of triangles|SSS congruence theorem]] for ' and QPB' to conclude that angles OPA' and OPB' are equal. Then use the [[Congruence (geometry)#Congruence of triangles|SAS congruence theorem]] for triangles OPA' and OPB' to conclude that angles POA and POB are equal.
 
==In a Cartesian coordinate system==
 
In a [[Cartesian coordinate system]], if two lines are each not parallel to either of the coordinate axes, then the two lines are perpendicular if and only if the product of their gradients is [[−1]].
 
==In relationship to parallel lines==
 
[[Image:perpendicular transversal v3.svg|right|thumb|200px|Lines ''a'' and ''b'' are parallel, as shown by the tick marks, and are cut by the [[transversal line]] c.]]
 
If two lines (''a'' and ''b'') are both perpendicular to a third line (''c''), all of the angles formed along the third line are right angles.  Therefore, in [[Euclidean geometry]], any two lines that are both perpendicular to a third line are parallel to each other, because of the [[parallel postulate]]. Conversely, if one line is perpendicular to a second line, it is also perpendicular to any line parallel to that second line.
 
In the figure at the right, all of the orange-shaded angles are congruent to each other and all of the green-shaded angles are congruent to each other, because [[vertical (angles)|vertical angles]] are congruent and alternate interior angles formed by a transversal cutting parallel lines are congruent. Therefore, if lines ''a'' and ''b'' are parallel, any of the following conclusions leads to all of the others:
 
* One of the angles in the diagram is a right angle.
* One of the orange-shaded angles is congruent to one of the green-shaded angles.
* Line 'c' is perpendicular to line 'a'.
* Line 'c' is perpendicular to line 'b'.
 
== Graph of functions ==
In the 2-dimensional plane, right angles can be formed by two intersected lines which the [[product (mathematics)|product]] of their [[slopes]] equals to −1. More precisely, defining two [[linear functions]]: {{nowrap|''y''<sub>1</sub> {{=}} ''a''<sub>1</sub>''x'' + ''b''<sub>1</sub>}} and {{nowrap|''y''<sub>2</sub> {{=}} ''a''<sub>2</sub>''x'' + ''b''<sub>2</sub>}}, the graph of the functions will be perpendicular and will make four right angles where the lines intersect if and only if {{nowrap|''a''<sub>1</sub>''a''<sub>2</sub> {{=}} −1}}. However, this method cannot be used if the slope is zero or undefined (the line is parallel to an axis).
 
For another method, let the two linear functions: ''a''<sub>1</sub>''x'' + ''b''<sub>1</sub>''y'' + ''c''<sub>1</sub> = 0 and ''a''<sub>2</sub>''x'' + ''b''<sub>2</sub>''y'' + ''c''<sub>2</sub> = 0. The lines will be perpendicular if and only if ''a''<sub>1</sub>''a''<sub>2</sub> + ''b''<sub>1</sub>''b''<sub>2</sub> = 0. This method is simplified from the [[dot product]] (or generally, [[inner product]]) of [[Euclidean vector|vector]]s. In particular, two vectors are considered orthogonal if their inner product is zero.
 
==See also==
 
* [[Orthogonality]]
* [[Parallel (geometry)]]
* [[Perpendicular component]] (of a vector)
* [[Surface normal]]
 
==External links==
{{wiktionary}}
*[http://www.mathopenref.com/perpendicular.html Definition: perpendicular] With interactive animation
*[http://www.mathopenref.com/constbisectline.html How to draw a perpendicular bisector of a line with compass and straight edge] Animated demonstration
*[http://www.mathopenref.com/constperpendray.html How to draw a perpendicular at the endpoint of a ray with compass and straight edge] Animated demonstration
 
[[Category:Elementary geometry]]
[[Category:Orientation]]

Latest revision as of 17:05, 2 December 2014

Before I left the inn, I said goodbye to Terrence and the rest of the staff, promising to return soon to this charming - and yes, quaint - bed-and-breakfast, the perfect place to ponder comic business or simply get comic relief in California's second largest city. In spite of the agreeable colors combination, they are tremendously opposed to over time. For this type of flooring, see my next cleaning solution. Granite countertops are best when they are installed by experts who know what they are doing in the granite business. To the masonry restoration engineer, understanding the evolution of mortar and cement are crucial to accurate and sound practice.

In the event you loved this informative article and you want to receive more information about www.heavensentfloorcare.com generously visit our own web-page. The basic, and ancient, lime and sand mortar, used for thousands of years, was still the primary unit masonry affixer and filler until the middle 19th century. Travertine tile is also more durable and resistant to chipping and cracking, although is it not invulnerable to the elements. All of the large and small projects are through with similar dedication. This has been made possible due to the modern cutting tools like water jet cutting, cnc machines, abrasive stones, bridge saws etc. In a masonry wall, this is usually accomplished by the mortar, not the masonry unit.

A wind turbine at a typical location in the Midwestern U. Masonry restoration of historic structures depends upon detailed knowledge of the chemical properties of mortar and cement ingredients, as well as their traditional manufacture and how that affected the finished material. Both of these preventative maintenance steps are crucial in protecting the natural beauty and value of your granite and yet they are so quick and simple to apply. Travertine is a durable natural stone that has been in use for centuries. It is possible to achieve much higher capacity factors by combining wind with a storage technology such as pumped hydro or compressed-air energy storage (CAES).

Wooden floors deals in engineered floors, laminatefloors, exterior floors, interior floors, designer floors, decking wood etc. " to these questions then now is your best time to remodel. ) You should have an average annual wind speed of 7 mph or greater, and the wind system tower should be high enough so the bottom of the turbine blades are 30 feet above any obstacle within 500 feet. You should always ensure that the service provider uses certified abrasives and polishes to bring natural and long lasting glow to your floors. One of the main disadvantages is that the real liability of loss of goods stored away remains with the customer.

October 1, the new museum will officially open, it will be a door leading to the history and culture, bringing us a research and discovery are known Liangzhu. For the right western bathroom decor you need to also include items such as tissue holders as well as waste baskets that make excellent cowboy accessories. You can call marble floor medallions as a type of mosaic art but in the modern days, it has proved to be much more than that. They taint surfaces and make them less attractive after some time, making your home experience less appealing. The market for the market restoration is a huge one as this method can well be employed to add in the original glow and strength to the marbles used in the residential or commercial spaces.

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