Answer: The Cartesian coordinate system is a mathematical tool used to locate points in a two-dimensional space. It was developed by the French mathematician and philosopher René Descartes in the 17th century and is named after him. The system consists of two perpendicular lines, known as the x-axis and y-axis, which intersect at a point called the origin.
The Cartesian coordinate system contains several key concepts and knowledge points. Firstly, it introduces the concept of coordinates, which are pairs of numbers that represent the position of a point in the plane. The x-coordinate represents the horizontal position of a point, while the y-coordinate represents the vertical position. The coordinates are written in the form (x, y), where x is the x-coordinate and y is the y-coordinate.
To determine the coordinates of a point in the Cartesian coordinate system, we follow a step-by-step process. Firstly, we locate the point on the x-axis by measuring the distance from the origin horizontally. If the point is to the right of the origin, the x-coordinate is positive; if it is to the left, the x-coordinate is negative. Next, we locate the point on the y-axis by measuring the distance from the origin vertically. If the point is above the origin, the y-coordinate is positive; if it is below, the y-coordinate is negative. Finally, we combine the x-coordinate and y-coordinate to form the coordinates of the point.
The formula or equation for the Cartesian coordinate system is simply the representation of a point in the form (x, y), where x and y are real numbers. There is no specific formula or equation for the Cartesian coordinate system itself, as it is a fundamental concept in mathematics.
To apply the Cartesian coordinate system, we can use the formula (x, y) to represent the coordinates of a point. For example, if we have a point P with coordinates (3, 4), it means that the point is located 3 units to the right of the origin and 4 units above the origin. We can also use the Cartesian coordinate system to solve various mathematical problems, such as finding the distance between two points or determining the equation of a line.
The symbol for the Cartesian coordinate system is the pair of parentheses ( ), which is used to enclose the coordinates of a point. For example, the point P with coordinates (3, 4) is represented using this symbol.
There are several methods for working with the Cartesian coordinate system. One common method is plotting points on a graph, where we mark the coordinates of a point on the x-axis and y-axis and then connect the points to form a shape or line. Another method is using the distance formula to find the distance between two points in the Cartesian plane. Additionally, we can use the slope-intercept form to determine the equation of a line in the Cartesian coordinate system.
Here are two solved examples on the Cartesian coordinate system:
Example 1: Find the coordinates of the point Q that is located 5 units to the left of the origin and 2 units below the origin.
Solution: Since the point is to the left of the origin, the x-coordinate is negative. Therefore, the x-coordinate of point Q is -5. Since the point is below the origin, the y-coordinate is also negative. Therefore, the y-coordinate of point Q is -2. The coordinates of point Q are (-5, -2).
Example 2: Determine the distance between the points A(2, 3) and B(5, -1).
Solution: We can use the distance formula to find the distance between two points in the Cartesian coordinate system. The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Substituting the coordinates of points A and B into the formula, we have:
d = √((5 - 2)^2 + (-1 - 3)^2) = √(3^2 + (-4)^2) = √(9 + 16) = √25 = 5
Therefore, the distance between points A and B is 5 units.
Now, let's move on to some practice problems on the Cartesian coordinate system:
Practice Problems:
Find the coordinates of the point R that is located 4 units to the right of the origin and 7 units above the origin.
Determine the distance between the points C(-2, 5) and D(3, -2).
Plot the points E(1, 2), F(-3, 4), and G(0, -1) on a graph.
Find the equation of the line passing through the points H(2, 3) and I(5, 1).
FAQ:
Question: What is the Cartesian coordinate system? Answer: The Cartesian coordinate system is a mathematical tool used to locate points in a two-dimensional space. It consists of two perpendicular lines, known as the x-axis and y-axis, which intersect at a point called the origin. The coordinates of a point in the Cartesian coordinate system are written in the form (x, y), where x is the x-coordinate and y is the y-coordinate.