Planes

Foundations of Data Science: Planes

We are setting up the foundation of Data Science with a brief introduction to Planes. This is the third article in the Data Science Foundation series.

We have learned about Linear Algebra and Coordinate Geometry in previous articles.

Planes

A point has no dimensions. A line is one-dimensional. A plane is two-dimensional (2D). A Solid is three-dimensional (3D).

In geometry, we can say a “plane” is a flat surface with no thickness. The coordinates specify the locations of points in a plane. A few higher-dimensional spaces can have planes as subspaces.

We have seen a Cartesian plane in the article on coordinate geometry. The Cartesian plane consists of two perpendicular lines whose intersection point is called the origin, and the origin is the zero point for both lines. The terms “X-axis” and “Y-axis” refer to the horizontal and vertical lines, respectively. The Cartesian plane coordinate point (x, y) indicates that the point’s horizontal and vertical distances from the origin are, respectively, x and y.

One Dimensional Space

A number line is a one-dimensional plane in the coordinate system. For the number line, draw a straight line and choose point 0 as the origin in the middle of the line. The line segment on the right of the origin is positive, whereas the line segment on the left is negative. A point can be drawn in a 1D plane with no size, i.e., no width, no length, and no depth.

number line

Figure 1.5: Number Line

In the number line in Figure 1.5, the value of the yellow point is 3.

In two-dimensional space, a line can be drawn, and a 2D plane can be drawn in 3D space.

Two-Dimensional  Space (2D space)

The 2D coordinate system is represented by the X-Y plane. The two mutually perpendicular lines represent the X-Y plane, as we have seen in the basics of coordinate geometry.

Line

Figure 1.6: 2D plane with a line

A line can be drawn in 2D space between points A and B. The distance formula can be used to determine a line’s length.

Distance between two points

 

 

Equation of a line

The equation of a line is:

y = ax + c

Here, x and y are the coordinates of points through which the line passes. ‘a’ is the slope of the line, and ‘c’ is the y-intercept of the line.

Three-Dimensional Space (3D Space)

In 3D space, there are three axes perpendicular to each other. For the earlier 2D space, there were x and y two axes perpendicular to each other; in 3D space, there is a third axis z, which is perpendicular to the xy plane. The three points (x, y, z) define any point on this plane. In this case, the positions along the x, y, and z axes are determined by the variables x, y, and z, respectively.

3D

 

 

 

 

 

 

 

 

 

Figure 1.7: Plane in 3D Space

The 3D plane has three axes (x, y, and z) in figure 1.7. The yellow point on the plane is three-dimensional and its value is (4,5,3)

Equation of the Plane

The equation of a plane is:

ax + by + cz = 0

We can say that a line in 2D space is equivalent to a plane in 3D space, and in n-dimensional space, it is called a hyperplane.

Stay Tuned!!

For system setup for data science with Python, click on the link below.

Data Science with Python: Introduction 

Keep learning and keep implementing!!

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