Problem 18 - Thomas Calculus

Question

In Exercises \(13-18\) , describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities. $$ (a)x=y, z=0 \quad \text { b. } x=y, \text { no restriction on } z $$

Step-by-Step Solution

Answer

(a) Line \( y=x \) on the xy-plane. (b) Plane \( x=y \) in space.

Step 1: Understand the Condition (a)

The condition \( x = y \) implies that the point lies on the plane where the x-coordinate and y-coordinate are equal. The condition \( z = 0 \) implies that the point lies in the plane \( z = 0 \), which is the xy-plane.

Step 2: Describe the Set for Condition (a)

The set of points that satisfy \( x = y, z = 0 \) is the line of intersection between the plane \( x = y \) and the xy-plane, which is the line \( y = x \) in the xy-plane.

Step 3: Understand the Condition (b)

The condition \( x = y \) again means the points lie on the plane where the x-coordinate equals the y-coordinate. 'No restriction on \( z \)' means that the points can have any z-coordinate value.

Step 4: Describe the Set for Condition (b)

The set of points that satisfy \( x = y \) with no restriction on \( z \), is the plane that passes through the line \( y = x \) for all values of \( z \). This is an infinite plane tilted such that for every point on the plane, \( x \) equals \( y \).

Key Concepts

Coordinate Planes

In three-dimensional geometry, coordinate planes are fundamental in determining the location of points in space. There are three primary coordinate planes:

• XY-plane: This plane is defined when the z-coordinate is zero. It extends infinitely in the x and y directions.
• YZ-plane: This plane occurs when the x-coordinate is zero, stretching along the y and z axes.
• XZ-plane: When the y-coordinate is zero, the xz-plane is formed, spreading across the x and z axes.

These planes help in visualizing points, lines, and figures in space, making it easier to compute distances and angles. Understanding where a point lies with respect to these planes is crucial. For instance, if a condition limits a point to the xy-plane, all its z-coordinate values must be zero.

Inequalities in Geometry

Inequalities in geometry describe the range or limits within which geometric figures exist. These constraints can shape a plane or space by limiting coordinates:

• Equality: An equation like \( x = y \) restricts points to those where the value of x equals y. This can describe a line, like the line y = x in the xy-plane.
• Inequality: More general expressions like \( x > y \) or \( z \geq 0 \) define broader regions, such as areas above or below a line or plane.

Understanding these constraints is key to visualizing the position and possible movement of points within given boundaries. They can describe planes, cylinders, or other shapes in space determined by these conditions.

Intersecting Planes

Intersecting planes occur when two or more planes meet. The intersection can result in different geometric configurations:

• Line of Intersection: When two planes intersect, the result is typically a line. For example, intersecting the plane \( x = y \) with the xy-plane results in the line \( y = x \), where all points join along this line.
• Point of Intersection: If three planes intersect at a single point, they will meet at one specific location that satisfies all equations.
• No Intersection: If planes are parallel, they never meet, thus having no point or line of intersection.

Intersecting planes are crucial in solving systems of equations as they help define the solution set, whether it's a point, line, or another geometric shape.

Three-Dimensional Space

Three-dimensional space or 3D space consists of three axes: x, y, and z. Each point in this space is defined by these three coordinates. Understanding 3D space enables us to:

• Visualize shapes and solids: Objects have length, width, and height.
• Determine distances and angles: 3D geometry involves calculations using distance formulas that account for three variables.
• Analyze vectors and transformations: Vectors help describe movement or direction, while transformations involve rotating or moving shapes.

Mastering three-dimensional space helps us navigate a world where most objects and experiences are not confined to just two dimensions. It forms the basis for engineering, architecture, physics, and many other fields.