Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the equation of the plane through \((2,5,1)\) that is parallel to the plane \(x-y+2 z=4\).

Step-by-Step Solution

Answer

The equation is \(x - y + 2z + 1 = 0\).

Step 1: Identify Given Plane Equation

The given plane equation is \(x-y+2z=4\). This plane is represented in the standard form \(Ax + By + Cz = D\), where \(A = 1\), \(B = -1\), and \(C = 2\). The vector \(\mathbf{n} = \langle 1, -1, 2 \rangle\) is the normal vector to this plane.

Step 2: Understand Parallel Planes

For two planes to be parallel, their normal vectors must be the same or scalar multiples of each other. Since the vector \(\mathbf{n} = \langle 1, -1, 2 \rangle\) is the normal vector of the given plane, it will also be the normal vector of the plane we want to find.

Step 3: Use Point-Plane Equation

To find the equation of a plane with a known normal vector passing through a specific point, we use the formula \(A(x-x_0) + B(y-y_0) + C(z-z_0) = 0\), where \((x_0, y_0, z_0)\) is the point \((2, 5, 1)\).

Step 4: Substitute Known Values

Substitute \(A=1\), \(B=-1\), \(C=2\), and \((x_0, y_0, z_0) = (2, 5, 1)\) into the equation: \[1(x-2) - 1(y-5) + 2(z-1) = 0\].

Step 5: Simplify the Equation

Expand and simplify the equation: \[1(x-2) - 1(y-5) + 2(z-1) = 0\] simplifies to \[x - 2 - y + 5 + 2z - 2 = 0\]. Combine like terms to get: \[x - y + 2z + 1 = 0\].

Key Concepts

Normal Vectors

A normal vector plays an important role in determining the orientation of a plane in 3D space. It is perpendicular to every possible direction that lies within the plane. Imagine holding a book upright in front of your face. The vector representing the edge of the book facing you (directly pointing out) is similar to a plane's normal vector.

When dealing with planes in mathematical or 3D geometric problems, the normal vector helps us to identify the direction in which the plane is "facing." A simple way to represent a normal vector in 3D is \(\mathbf{n} = \langle A, B, C \rangle\), where A, B, and C are the coefficients of x, y, and z in the plane equation. In this context, for the plane described by \(x-y+2z=4\), the normal vector is \(\mathbf{n} = \langle 1, -1, 2 \rangle\).

Understanding the normal vector is pivotal, as it aids in identifying whether two planes are parallel and assists in the calculation of the equation of a new plane.

Parallel Planes

Parallel planes have the unique characteristic of having identical or proportionate normal vectors. This means that if the normal vector of one plane is \(\mathbf{n} = \langle A, B, C \rangle\), any plane parallel to it will either have the same normal vector or a scalar multiple like \(k\langle A, B, C \rangle\), where k is a constant.

Because of this property, determining if two planes are parallel becomes straightforward. You need to only compare their normal vectors. In our example, since the existing plane has a normal vector of \(\langle 1, -1, 2 \rangle\), any plane parallel to it will have this same normal vector.

Identifying parallel planes is useful in constructing geometric solutions and can simplify 3D modeling tasks such as defining bounding surfaces or working within parallel lines in graphical representations.

Equation of a Plane

The equation of a plane in 3D, generally written as \(Ax + By + Cz = D\), provides a way to describe all points that lie on the plane. This form reveals not just the orientation but also the specific spatial positioning of the plane.

To find the equation of a plane passing through a specified point with a given normal vector, the point-plane equation \(A(x-x_0) + B(y-y_0) + C(z-z_0) = 0\) is used. The point \(\left(x_0, y_0, z_0\right)\) is any known point on the plane. By substituting the normal vector components and the point coordinates into this formula, you derive the plane's equation.

For example, substituting into this formula using \(\langle 1, -1, 2 \rangle\) as the normal vector and (2, 5, 1) as the point yields the equation \(x - y + 2z + 1 = 0\). This clearly describes another plane parallel to the given plane through a specified point.

3D Geometry

Three-dimensional geometry extends beyond simple 2D shapes to encompass depth, width, and height. It involves understanding how objects occupy space and relate to each other in a 3D coordinate system.

Basic elements in 3D geometry include points, lines, and planes. Together, these create the frameworks for more complex structures like prisms, spheres, and polyhedra. The Cartesian coordinate system is most often used to define locations in this space, with each point determined by three coordinates: \(x, y, z\).

In particular, plane equations that describe surfaces in 3D are fundamental to modeling and computational graphics. By setting conditions like parallelism or perpendicularity, we can manipulate and create complex 3D forms for various applications, from computer graphics to solutions in physics and engineering.