Problem 14 - Precalculus with Calculus Previews

Question

Determine an equation of a plane parallel to a coordinate plane that contains the given pair of points. (a) (3,4,-5),(-2,8,-5) (b) (1,-1,1),(1,-1,-1) (c) (-2,1,2),(2,4,2)

Step-by-Step Solution

Answer

(a) \(z = -5\); (b) \(x = 1\); (c) \(z = 2\).

Step 1: Identifying the Coordinate of Constancy (Part a)

The given points are \((3,4,-5)\) and \((-2,8,-5)\). Here, both points share the same \(z\)-coordinate, \(z = -5\). This means the plane containing these points is parallel to the \(xy\)-plane and has an equation of the form \(z = \text{constant}\).

Step 2: Equation of the Plane (Part a)

Since the \(z\)-coordinate is constant for both points, the equation of the plane is \(z = -5\). This equation represents a horizontal plane parallel to the \(xy\)-plane at \(z = -5\).

Step 3: Identifying the Coordinate of Constancy (Part b)

The given points are \((1,-1,1)\) and \((1,-1,-1)\). Here, both points share the same \(x\)- and \(y\)-coordinates, \(x = 1\) and \(y = -1\). This means the plane containing these points is parallel to the \(yz\)-plane and has an equation of the form \(x = \text{constant}\).

Step 4: Equation of the Plane (Part b)

Since the \(x\)-coordinate is constant for both points, the equation of the plane is \(x = 1\). This equation represents a vertical plane parallel to the \(yz\)-plane at \(x = 1\).

Step 5: Identifying the Coordinate of Constancy (Part c)

The given points are \((-2,1,2)\) and \((2,4,2)\). Here, both points share the same \(z\)-coordinate, \(z = 2\). This indicates that the plane containing these points is parallel to the \(xy\)-plane and has an equation of the form \(z = \text{constant}\).

Step 6: Equation of the Plane (Part c)

Since the \(z\)-coordinate is constant for both points, the equation of the plane is \(z = 2\). This equation describes a horizontal plane parallel to the \(xy\)-plane at \(z = 2\).

Key Concepts

Coordinate Plane

The coordinate plane is a fundamental concept in geometry and algebra, representing a two-dimensional flat surface. It is defined by two perpendicular lines intersecting at a point called the origin. These lines are called the x-axis (horizontal) and y-axis (vertical). The coordinate plane allows us to express points using a pair of numbers known as coordinates, typically written as \((x, y)\). These coordinates specify the position of a point in respect to the axes. When considering three-dimensional space, we introduce a third axis, the z-axis, perpendicular to the x- and y-axes. This transforms the two-dimensional coordinate plane into a three-dimensional coordinate system, where each point is expressed as \((x, y, z)\). This 3D space is crucial when talking about planes, vectors, and many geometric shapes.

Parallel Planes

Parallel planes are two or more planes that do not intersect at any point. In three-dimensional geometry, these planes remain equidistant from each other no matter how far they extend. The concept of parallel planes is similar to parallel lines but applied in a 3D context.

• Properties: Parallel planes must maintain a constant distance, so if you have two planes expressed as \(Ax + By + Cz = D_1\) and \(Ax + By + Cz = D_2\), they are parallel as they share the same normal vector \((A, B, C)\).
• Example: In the context of the problem, the planes defined in parts (a) and (c) are parallel to the \(xy\)-plane, as their equations \(z = -5\) and \(z = 2\) indicate. These planes will never meet the \(xy\)-plane or each other as they are parallel and horizontal.

This idea is essential for understanding how geometric objects relate to one another in space.

Plane Formulation

Formulating a plane in space involves determining an equation that defines its infinite flatness. A plane can be uniquely specified using various forms, one of the simplest being when it is defined in terms of a constant coordinate, as seen in the solution steps for the exercise. This can occur when:

• All points on the plane share identical \(x\)-coordinates, leading to an equation of form \(x = c\).
• All points have the same \(y\)-coordinates, giving an equation like \(y = c\).
• All points exhibit the same \(z\)-coordinates, resulting in \(z = c\).

For instance, in the exercise, determining which coordinate remains constant (or unchanged among given points) allowed the formulation of the plane equations, as shown in each part of the example.

Plane Equations

Plane equations are mathematical expressions that define a flat surface extending infinitely in three-dimensional space. A common form of the equation of a plane is \(Ax + By + Cz = D\), where \(A, B,\) and \(C\) are coefficients representing the plane's orientation and \(D\) is a constant. This equation describes a plane based on its relationship to the origin and axes, influenced directly by the normal vector \((A, B, C)\).
In simpler cases, like the exercise you worked on, recognizing that a plane has a constant value for a specific coordinate simplifies the plane's equation to \(x =\) constant, \(y =\) constant, or\(z =\) constant. These straightforward equations demonstrate planes parallel to one of the major coordinate planes, offering an easily understandable perspective of spatial relationships.