Final answer:
To find the distance between two Cartesian points, use the distance formula derived from the Pythagorean theorem. To convert Cartesian coordinates to polar coordinates, calculate the radial distance r and the angle θ with the x-axis. The Cartesian system can also be extended to three-dimensional space by adding a z-coordinate.
Step-by-step explanation:
The coordinates of a point in the Cartesian coordinate system are the numerical addresses that specify its exact location on a two-dimensional graph. In the Cartesian system, the horizontal axis is generally labeled as the x-axis, and the vertical axis is labeled as the y-axis. To find the distance between two points in the Cartesian plane, we use the distance formula which is derived from the Pythagorean theorem. Given two points A(x1, y1) and B(x2, y2), the distance d between them is calculated as:
d = √((x2 - x1)² + (y2 - y1)²)
For converting Cartesian coordinates to polar coordinates, we use the following equations:
For example, to find the polar coordinates of point A(2.00 m, -4.00 m), we calculate:
If the point is in a different quadrant, the angle θ would be adjusted accordingly based on the signs of x and y to represent the correct direction. For point B(-3.00 m, 3.00 m), we would similarly find its polar coordinates.
The Cartesian coordinate system also extends to three-dimensional space by incorporating a third coordinate, z, for height or depth.
Understanding these concepts is critical for fields like physics, engineering, and computer graphics, where spatial relationships and movement are key.