Problem 19 - Thomas Calculus

Question

In Exercises \(19-28\) , describe the given set with a single equation or with a pair of equations. The plane perpendicular to the a. \(x\) -axis at \((3,0,0) \quad\) b. \(y\) -axis at \((0,-1,0)\) c. \(z\) -axis at \((0,0,-2)\)

Step-by-Step Solution

Answer

a) \(x = 3\), b) \(y = -1\), c) \(z = -2\).

Step 1: Understanding the Problem

We need to find equations that describe planes perpendicular to the coordinate axes, passing through specific points given for each axis.

Step 2: Describing the Plane Perpendicular to the x-axis

For part (a), the plane is perpendicular to the x-axis at the point (3,0,0). A plane perpendicular to the x-axis has an equation of the form \(x = c\), where \(c\) is the specific x-coordinate. Thus, the equation is \(x = 3\).

Step 3: Describing the Plane Perpendicular to the y-axis

For part (b), the plane is perpendicular to the y-axis at the point (0,-1,0). A plane perpendicular to the y-axis is described by the equation \(y = c\), where \(c\) is the y-coordinate. Therefore, the equation is \(y = -1\).

Step 4: Describing the Plane Perpendicular to the z-axis

For part (c), the plane is perpendicular to the z-axis at the point (0,0,-2). A plane perpendicular to the z-axis has an equation of \(z = c\), where \(c\) is the z-coordinate. The equation is \(z = -2\).

Key Concepts

Understanding Coordinate Axes

Coordinate axes in three-dimensional geometry refer to the three lines that define the

• horizontal (x-axis),
• vertical (y-axis), and
• depth (z-axis)

space of a coordinate system.
These axes act as reference points for locating positions in the space. In three-dimensional space, any point can be determined by its coordinates

• The x-coordinate represents the distance along the x-axis
• The y-coordinate represents the distance along the y-axis,
• and the z-coordinate indicates the distance along the z-axis.

The origin, represented as (0,0,0), is the central point where all three axes intersect.
Understanding these axes is crucial for describing planes and directions in three-dimensional space.

Exploring Perpendicular Planes

A plane is said to be perpendicular to an axis if it intersects the axis at a right angle, meaning the angles formed are all right angles (90°) along that axis.
In our exercise, describing these planes involves identifying which coordinate remains constant while the others can vary.When a plane is perpendicular to:

• The x-axis at some point (x, 0, 0), it corresponds to an equation of the form \( x = c \), where \( c \) is the x-coordinate.
• Similarly, if it is perpendicular to the y-axis at (0, y, 0), the equation is \( y = c \).
• And if it is perpendicular to the z-axis at (0, 0, z), the equation becomes \( z = c \).

These equations signify that a plane passes perpendicularly through the given point and extends infinitely along the other two axes.
This understanding is essential for visualizing and solving problems in three-dimensional geometry.

Navigating Three-Dimensional Geometry

Three-dimensional geometry involves understanding objects and spaces having three dimensions: length, width, and height.
It goes beyond the two-dimensional geometry of shapes like squares and circles that we learn early on. In three-dimensional geometry:

• We encounter solids such as spheres, cones, and cubes.
• We also deal with planes, lines, and points in a three-dimensional space.

Visualizing these objects requires imagining or drawing them in a manner that reflects their depth.
Knowing how to describe and define planes, such as those that are perpendicular to an axis, is vital for creating correct models of structures and solving geometrical problems. For instance, by understanding the equation of a plane, we can identify how that plane segments or interacts with other parts of the space.
This helps in fields like engineering and physics where precise modeling is needed for constructions and understanding forces.