Analytic Geometry

Analytics Geometry of Two and Three Dimensions

Analytic Geometry is an invention of Descartes and Fermat. It is also referred to as coordinate geometry. It is a mathematical subject where algebraic symbolism and methods are used to represent and solve geometry problems.

Analytic geometry basically establishes a connection between geometric curves and algebraic equations. It is often referred to as a branch of algebra that is used to model geometric objects. It helps in the reformulation of algebraic problems in geometry and vice versa, where the methods and principles of both fields can be used to solve the problems in the other.

Analytical geometry of three dimensions forms the foundation of various modern fields of geometry that include algebraic, differential, discrete, and computational geometry. The application of Analytic geometry spreads across a wide range of fields including physics and engineering, and also in aviation, rocketry, space science, and spaceflight.

Trigonometry and analytic geometry encompass some very important subjects such as midpoints and distance, parallel and perpendicular lines on the coordinate plane, dividing line segments, distance between the line and a point, etc. This field is often deemed as the next level of mathematics, that helps in the field of learning logical thinking and improving problem-solving skills of an individual.
In this article, we will discuss the most important aspects of analytic geometry including formulas, cartesian plane, analytic geometry in three dimensions, its applications, and some solved sums for a better and more in depth understanding.

What is Analytic Geometry?

In analytic geometry, points are defined as ordered pairs of numbers, for instance, (x, y), and the straight lines are consequently defined as the sets of points that satisfy linear equations. 

Analytic geometry defines geometric axioms as derivable theorems. 

Here is an example, for any two distinct points, say (x1, y1) and (x2, y2), there is a single line ax + by + c = 0 that passes through these points. The coefficients (a, b, c) can be found (up to a constant factor) from a linear system of two equations:

ax1 + by1 + c = 0

ax2 + by2 + c = 0,

or directly from the determinant equation. It is a contradiction to symmetric geometry, where no candidates or formulas are used.

Both two dimensional and three-dimensional geometry are used in coordinate geometry. The most common terms used in analytic geometry are:

• Planes

• Coordinates

Planes

For a clear understanding of analytical geometry of three dimensions or two, and the importance and applications of analytic geometry, we need to be clear about the definition of a plane. A plane is a flat surface that continues infinitely in both directions. Analytic Geometry helps locate any point on this plane with the coordinates of the point in the X and Y plane.

Coordinates

Two ordered pairs which define the location of any given point in a plane can be defined as coordinates. The box below will help us get a better understanding.

In this grid, the location of letter x is B2 i.e. Column B and row 2. So, B and 2 are the coordinates of this box, x.

Analytical Geometry of three dimensions has different kinds of coordinates. The most important ones are:

Cartesian Coordinates

The Cartesian coordinate is the most well-known coordinate system in which every point has an x-coordinate and y-coordinate which represents its horizontal position, and vertical position respectively.

Polar Coordinates

A coordinate system where each point in a plane is denoted by the distance ‘r’ from the origin and the angle θ from the polar axis is known as the polar coordinates.

Cylindrical Coordinates

In cylindrical coordinates, all the points are expressed by their height, radius from z-axis, and the angle projected on the xy-plane with respect to the horizontal axis. The height, radius, and angle are denoted by h, r, and θ, respectively.

Spherical Coordinates

In this type of coordinates the point in space is denoted by its distance from the origin ( ρ), the angle projected on the xy-plane with respect to the horizontal axis (θ) and also another angle, with respect to the z-axis (φ).

Did you Know?

1. The origin is the point of intersection of the axis(x-axis and y-axis). Both x and y-axis are zero at this point.

2. x-axis – The values at the right-hand side of the x-axis are positive and ones on the left-hand side are negative.

3. y-axis – The values above the origin are positive and the ones below the origin are negative.

4. You need two numbers to locate a plane in the order of writing the location of the X-axis first and Y-axis after that. This denotes a unique position on the plane.

Analytic Geometry Formulas

To find measurements of geometric figures, formulas for analytic geometry are extremely useful. Graphs and coordinates are the vast fields of application of analytic geometry. Application of analytic geometry in the engineering field and sciences involves the study of the rate of change in varying quantities and helps to show the relationship between the quantities involved. Check out the analytic geometry formulas here.

Distance Formula

Let the two points be A and B, having coordinates to be (x1,y1) and (x2,y2) respectively.

Thus, the distance between two points is given as-

d = √[(x2-x1)2+(y2-y1)2]

Midpoint Theorem Formula

Let A and B are some points in a plane, which is joined to form a line, having coordinates (x1,y1) and (x2,y2), respectively. Suppose, M(x,y) is the midpoint of the line connecting the point A and B then its formula is given by;

M(x,y) = [(x1+x2/2),(y1+y2/2)]

Angle Formula

Let two lines have slope = m1 and m2, and θ be the angle formed between the two lines, A and B. This is represented as;

tan θ = m1-m2/1+m1m2

Solved Examples

How to find the distance between two points A and B such that the coordinates of A and B are (5, -3) and (2, 1)?

Given that, the coordinates are:

• A = (5, -3)  = (x1, y1)

• B = (2, 1) = (x2,y2)

The formula to find the distance between two points is given as:

• Distance,d = √[(x2-x1)2+(y2-y1)2]

• d = √[(2-5)2+(1- (-3))2]

• d =√[(-3)2+(4)2]

• d =√[9+16]

• d =√(25)

• d = 5

Thus, the distance between two points A and B is 5.