Rectangular coordinates, also known as Cartesian coordinates, are a system used to locate points in a two-dimensional or three-dimensional space. This coordinate system uses perpendicular axes, typically labeled x, y, and sometimes z, to represent the position of a point relative to a fixed origin.
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In a two-dimensional rectangular coordinate system, the x-axis represents the horizontal position, and the y-axis represents the vertical position.
The coordinates of a point in a two-dimensional rectangular coordinate system are given as an ordered pair (x, y), where x is the horizontal position, and y is the vertical position.
In a three-dimensional rectangular coordinate system, the z-axis is added to represent the depth or elevation, and the coordinates of a point are given as an ordered triple (x, y, z).
Rectangular coordinates are useful for representing and analyzing the motion of objects, such as in the study of vectors and their applications.
The distance between two points in a rectangular coordinate system can be calculated using the Pythagorean theorem.
Review Questions
Explain how rectangular coordinates are used to represent the position of a point in a two-dimensional space.
In a two-dimensional rectangular coordinate system, the position of a point is represented by an ordered pair (x, y), where x represents the horizontal position and y represents the vertical position. The point is located at the intersection of the x-axis and the y-axis, with the x-coordinate indicating the distance from the origin along the horizontal axis, and the y-coordinate indicating the distance from the origin along the vertical axis. This system allows for the precise representation and analysis of the location of objects or points in a two-dimensional plane.
Describe how rectangular coordinates are used to represent the motion of an object in the context of vectors.
Rectangular coordinates are essential in the study of vectors, which are quantities that have both magnitude and direction. Vectors can be represented in a rectangular coordinate system, where the x, y, and sometimes z components of the vector correspond to the distances along the respective axes. This allows for the decomposition of a vector into its components, which can be used to analyze the motion and displacement of an object. The rectangular coordinate system provides a convenient way to describe the direction and magnitude of a vector, which is crucial in the study of physics, engineering, and other fields that involve the analysis of motion.
Evaluate the advantages of using rectangular coordinates over polar coordinates for certain applications, particularly in the context of 8.8 Vectors.
While both rectangular and polar coordinate systems have their applications, rectangular coordinates are often preferred in the context of 8.8 Vectors. Rectangular coordinates provide a more intuitive and straightforward way to represent and analyze the components of a vector, which is essential for understanding vector operations such as addition, subtraction, and scalar multiplication. Additionally, the Cartesian nature of rectangular coordinates makes it easier to perform calculations and visualize the motion of objects, as the x, y, and sometimes z components can be treated independently. This is particularly useful when working with vector-based problems in physics, engineering, and other fields that involve the analysis of motion and displacement, which are central topics in the 8.8 Vectors section.
A coordinate system that uses the distance from the origin (radius) and the angle from a reference direction (angle) to specify the location of a point.