Cartesian System

The Cartesian system is a fundamental concept in coordinate geometry. It was developed by French mathematician René Descartes in the 17th century and revolutionized mathematics by introducing a way to describe the position of points in the plane through numerical coordinates. This system allows us to represent geometric shapes algebraically and solve geometric problems numerically. In this explanation, we will explore the dimensions of the Cartesian system, how to interpret graphs on the Cartesian plane, how to plot points, and more.

Understanding the basics

The Cartesian system uses two perpendicular lines called axes to divide a plane into four regions. These axes are typically labeled as the x-axis (horizontal) and the y-axis (vertical). The point where these two axes intersect is known as the origin, typically represented as (0, 0).

Axis and quaternions

The lines that form the axis divide the plane into four parts, called quadrants. They are numbered counterclockwise:

• Quadrant I: Both x and y coordinates are positive (top right).
• Quadrant II: x is negative, y is positive (top left).
• Quadrant III: Both x and y coordinates are negative (bottom left).
• Fourth quadrant: x is positive, y is negative (bottom right).

Drawing points on the Cartesian plane

Every point on the Cartesian plane can be identified by an ordered pair (x, y), where x is the value on the x-axis, and y is the value on the y-axis. Let's plot some points to see how this works.

The above figure shows four points marked on the Cartesian plane:

• The red dot (4, 4) is in quadrant I.
• The green point (-4, 4) is in quadrant II.
• The blue point (-4, -4) is in quadrant III.
• The purple point (4, -4) is in the fourth quadrant.

Distance between two points

To find the distance between two points on the Cartesian plane, you can use the distance formula. If you have two points P(x1, y1) and Q(x2, y2), the distance d between them is given by:

d = √((x2 - x1)² + (y2 - y1)²)

Let's apply this formula to find the distance between the points (4, 4) and (-4, 4) from our previous example:

(x1, y1) = (4, 4) (x2, y2) = (-4, 4) d = √((-4 - 4)² + (4 - 4)²) d = √((-8)² + (0)²) d = √(64 + 0) d = √64 d = 8

Therefore, the distance between these two points is 8 units.

Midpoint formula

The midpoint of a line segment is the point that divides the segment into two equal parts. The midpoint M of the segment joining two points P(x1, y1) and Q(x2, y2) can be found using the midpoint formula:

M = ((x1 + x2)/2, (y1 + y2)/2)

Let us find the midpoint between the points (4, 4) and (-4, 4):

M = ((4 + (-4))/2, (4 + 4)/2) M = (0/2, 8/2) M = (0, 4)

Thus, the midpoint is at (0, 4).

Slope of the line

The slope of a line is a measure of its steepness. In the Cartesian plane, the slope m between two points (x1, y1) and (x2, y2) is calculated as follows:

m = (y2 - y1) / (x2 - x1)

This formula is derived by dividing the change in y values by the change in x values, also known as 'run over rise'. Using the points (4, 4) and (-4, 4), let's calculate the slope:

m = (4 - 4) / (-4 - 4) m = 0 / -8 m = 0

This result indicates that the line is horizontal.

Equations of lines

Slope intercept form

A common way to express linear equations in a Cartesian system is in slope-intercept form:

y = mx + b

where m is the slope and b is the y-intercept (the point where the line intersects the y-axis).

Point-slope form

The second form is the point-slope form, which is useful when you know the slope and a particular point on the line:

y - y1 = m(x - x1)

Here, (x1, y1) is a known point on the line and m is the slope.

General form

A linear equation can also be written in general form:

Ax + By + C = 0

This form represents a line for any values of A, B, and C, provided that all are not zero.

Application

The Cartesian system has many applications in various fields. It contributes greatly to fields such as physics, engineering, computer graphics, navigation and much more by providing a systematic approach to understanding space and looking at problems algebraically. It allows graphs to be plotted and interpreted and is helpful in solving equations visually.

Examples and exercises

To further strengthen the understanding, let us consider some practical examples of points, lines, and equations in the Cartesian plane.

Example 1: Finding the equation of a line

Given points (2, 3) and (4, 7), find the equation of the line passing through these points.

First, use the slope formula:

m = (7 - 3) / (4 - 2) m = 4 / 2 m = 2

Now, use the point-gradient formula with the point (2, 3):

y - 3 = 2(x - 2) y - 3 = 2x - 4 y = 2x - 1

The equation of the line is y = 2x - 1.

Example 2: Identifying points on a line

Check whether the point (1, 1) lies on the line y = 2x - 1.

Substitute x = 1 into the equation:

y = 2(1) - 1 y = 2 - 1 y = 1

Since y = 1 corresponds to the y-coordinate of the point, (1, 1) is on the line.

Conclusion

The Cartesian system is an indispensable tool in mathematics, providing a framework for translating geometric problems into algebraic contexts. Mastering this system is foundational to further study in mathematics and its applications in real-world contexts.

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