Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width. Equations of Planes In the first section of this chapter we saw a couple of equations of planes.
To avoid this vicious circle certain concepts must be taken as primitive concepts; terms which are given no definition. When the line concept is a primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy. In a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with.
In this circumstance it is possible that a description or mental image of a primitive notion is provided to give a foundation to build the notion on which would formally be based on the unstated axioms.
Descriptions of this type may be referred to, by some authors, as definitions in this informal style of presentation. These are not true definitions and could not be used in formal proofs of statements.
The "definition" of line in Euclid's Elements falls into this category. In Euclidean geometry[ edit ] See also: Euclidean geometry When geometry was first formalised by Euclid in the Elementshe defined a general line straight or curved to be "breadthless length" with a straight line being a line "which lies evenly with the points on itself".
In fact, Euclid did not use these definitions in this work and probably included them just to make it clear to the reader what was being discussed. In modern geometry, a line is simply taken as an undefined object with properties given by axioms but is sometimes defined as a set of points obeying a linear relationship when some other fundamental concept is left undefined.
In an axiomatic formulation of Euclidean geometry, such as that of Hilbert Euclid's original axioms contained various flaws which have been corrected by modern mathematicians a line is stated to have certain properties which relate it to other lines and points.
For example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect in at most one point.
In higher dimensions, two lines that do not intersect are parallel if they are contained in a planeor skew if they are not. Any collection of finitely many lines partitions the plane into convex polygons possibly unbounded ; this partition is known as an arrangement of lines.
On the Cartesian plane[ edit ] Lines in a Cartesian plane or, more generally, in affine coordinatescan be described algebraically by linear equations. In two dimensionsthe equation for non-vertical lines is often given in the slope-intercept form:YOUR TURN: Find the equation of the line passing through the points (-4, 5) and (2, -3).
Algebra > Lines > Perpendicular Lines > Perpendicular Lines Cruncher Perpendicular Lines Cruncher This Algebra Cruncher generates an endless number of practice problems for finding the equation of the line that is perpendicular to a given line and that passes through a given point -- with hints and solutions!
perpendicular use the opposite-reciprocal slope). Then use point-slope form. Example: Write the equation of a line in slope-intercept form that is perpendicular to 2x-3y = 6 and goes through. After completing this tutorial, you should be able to: Find the slope given a graph, two points or an equation.
Write a linear equation in slope/intercept form. In the last lesson, I showed you how to get the equation of a line given a point and a slope using the formula.
Anytime we need to get the equation of a line, we need two things. Simply knowing how to take a linear equation and graph it is only half of the battle. You should also be able to come up with the equation if you're given the right information.