Question

Describe each surface given by the equations \(x=a\) \(y=b,\) and \(z=c\)

Short answer

The equation \(x=a\) describes a plane parallel to the yz-plane. The equation \(y=b\) describes a plane parallel to the xz-plane. The equation \(z=c\) describes a plane parallel to the xy-plane.

Step-by-step solution

Analyzing the Equation x=a

The equation \(x=a\) describes a plane which is parallel to the yz-plane (plane formed by y-axis and z-axis). This is because no matter what the values of y and z are, the value of x remains constant and equals to a.

Analyzing the Equation y=b

The equation \(y=b\) describes a plane which is parallel to the xz-plane (plane formed by x-axis and z-axis). This is because no matter what the values of x and z are, the value of y remains constant and equals to b.

Analyzing the Equation z=c

The equation \(z=c\) describes a plane which is parallel to the xy-plane (plane formed by x-axis and y-axis). This is because no matter what the values of x and y are, the value of z remains constant and equals to c.

Key concepts

Plane Equations

Understanding the equations of planes is fundamental when dealing with three-dimensional geometry. A plane equation in three dimensions typically takes the form \(Ax + By + Cz + D = 0\), where \(A\), \(B\), and \(C\) represent the coefficients that provide the direction of the normal to the plane, and \(D\) is a constant that determines the plane's position relative to the origin. When we look at simple equations like \(x=a\), \(y=b\), or \(z=c\), we are actually describing planes that are parallel to the main coordinate planes.

For instance, \(x=a\) is a vertical plane that intersects the x-axis at \(a\) and is parallel to the yz-plane, essentially regardless of the values that \(y\) and \(z\) might take. This is a visual and conceptual shortcut that helps students understand more complex plane equations they might encounter.

Parallel Planes

Parallel planes share the fascinating trait of never meeting, no matter how far they extend in space. This concept is important in geometry and is visually illustrated by planes such as those described by the equations \(x=a\), \(y=b\), and \(z=c\).

Imagine two planes in three-dimensional space, both defined by \(x=a\) but with different \(a\) values, say \(x=1\) and \(x=3\). These two planes will not intersect and are perfect examples of parallel planes because they have the same orientation but are at different positions along the x-axis.
Understanding parallelism is crucial in fields like engineering and physics, where it is used to solve problems involving object alignments, forces, and motions in three-dimensional spaces.

Three-Dimensional Coordinates

Grasping the idea of three-dimensional coordinates is like unlocking a new level of understanding space. It's the transition from the flat world of two-dimensional geometry into the vast expanse of 3D space. Each point in this space is described by three coordinates \(x, y, z\), representing its position along the x-axis, y-axis, and z-axis respectively.

The beauty of three-dimensional coordinates is that they allow us to pinpoint the location of any point or object in space precisely. This ties back to our plane equations \(x=a\), \(y=b\), and \(z=c\), where \(a\), \(b\), and \(c\) are specific coordinates along each axis, indicating where the plane intersects that respective axis. Mastering this concept is essential for students as it forms the basis of many advanced topics in mathematics, physics, and engineering.