The xy-plane, also known as the horizontal plane, is a fundamental concept in three-dimensional Cartesian coordinate systems. It is one of the three principal planes that intersect at the origin, forming a three-dimensional coordinate system. The xy-plane is defined as the plane that is perpendicular to the z-axis and contains the x and y axes.
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The xy-plane is the fundamental plane in three-dimensional space, and it is the primary plane used for visualizing and analyzing functions of several variables.
Vectors in three-dimensional space can be represented in the xy-plane, with their components along the x and y axes.
Cylindrical and spherical coordinate systems both use the xy-plane as the fundamental plane for their coordinate systems.
The xy-plane is often used to graph and analyze functions of two variables, where the independent variables are the x and y coordinates.
The orientation of the xy-plane is such that the positive x-axis points to the right, the positive y-axis points upward, and the positive z-axis points out of the plane towards the viewer.
Review Questions
Explain how the xy-plane is used in the context of vectors in three dimensions.
In the study of vectors in three dimensions (Section 2.2), the xy-plane is used to represent the components of a vector along the x and y axes. A vector in three-dimensional space can be decomposed into its $x$, $y$, and $z$ components, with the $x$ and $y$ components lying within the xy-plane. This allows for the visualization and analysis of vector operations, such as addition and scalar multiplication, within the xy-plane.
Describe the role of the xy-plane in the context of cylindrical and spherical coordinate systems (Section 2.7).
In the study of cylindrical and spherical coordinate systems (Section 2.7), the xy-plane serves as the fundamental plane. In the cylindrical coordinate system, the xy-plane represents the base plane, with the $\rho$ (radial) coordinate specifying the distance from the origin to the projection of a point onto the xy-plane, and the $\theta$ (angular) coordinate specifying the angle between the positive $x$-axis and the projection of the point onto the xy-plane. Similarly, in the spherical coordinate system, the xy-plane is the reference plane, with the $\phi$ (azimuthal) angle measured from the positive $x$-axis in the xy-plane.
Discuss the importance of the xy-plane in the context of functions of several variables (Section 4.1).
In the study of functions of several variables (Section 4.1), the xy-plane is the primary plane used for visualizing and analyzing these functions. A function of two variables, $f(x, y)$, can be represented as a surface in three-dimensional space, with the $x$ and $y$ coordinates defining the position on the xy-plane and the $z$ coordinate representing the function value. This geometric interpretation of functions of several variables allows for the exploration of properties such as level curves, partial derivatives, and gradients, all of which are closely tied to the xy-plane.
Related terms
Cartesian Coordinate System: A coordinate system that specifies the location of a point in space using three mutually perpendicular axes: the x-axis, y-axis, and z-axis.