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Chapter 2: Problem 354

\(\begin{array}{llll} & \text { Show } & \text { that } & \text { quadric } & \text { surface }\end{array}\) \(x^{2}+y^{2}+z^{2}-2 x y-2 x z+2 y z-1=0\) reduces to two parallel planes passing.

Short Answer

Expert verified
The quadric reduces to two parallel planes: \(x - y = 1\) and \(y - z = -1\).

Step by step solution

01

Reorganize the equation

Start by observing the given equation: \( x^{2}+y^{2}+z^{2}-2xy-2xz+2yz-1=0 \). The quadratic terms need to be reorganized to simplify the expression, possibly into a more recognizable form or through factorization.
02

Complete the square

Identify terms that allow the completion of the square. Combine and rearrange terms to facilitate this. Split the quadratic terms as follows: \( (x-y)^2 + (y-z)^2 + 2(yz - 1) = 0 \). Expanding them results in: \( x^2 + y^2 - 2xy + y^2 + z^2 - 2yz + 2yz - 2 =0 \), which simplifies the initial equation to \( (x-y)^2 + (y-z)^2 = 1 \). The equation \( 2(yz - 1) \) transforms the offset.
03

Analyze for factorable form

Equate each squared term such that the expression points to parallel forms. Evaluating \( (x-y)^2 + (y-z)^2 = 1 \) suggests parallel planes conceptually. Recognize that translating and recomposing parts into forms \( (x-y) - 1=0 \) and \( (y-z) + 1=0 \) represent planes symmetric about the axis.
04

Rewrite as Plane Equations

Reformulate the simplified form into explicitly two plane equations. The resulting planes are expressed in their explicit forms: \( x - y = 1 \) and \( y - z = -1 \). These describe two distinct planes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parallel Planes
Parallel planes are surfaces in 3D space that never intersect. These planes maintain a constant distance from each other. In mathematical terms, parallel planes have normal vectors that point in the same direction.
For the given exercise, our goal is to show that the complex quadric surface simplifies into two parallel planes. Once we rearrange and manipulate the equation, it boils down to simpler plane equations, confirming their parallel nature.
Seeing these planes emerge in their parallel configuration allows you to understand the elegance in geometrically complex systems, validating how transformations leads to simpler interpretations.
Complete the Square
Completing the square is a technique often used to make quadratic expressions easier to handle. It involves rewriting an imperfect square into a perfect square, which reveals hidden structures within the equation.
In this exercise, this method helps to uncover the true nature of the given quadric surface. By grouping terms such as \( (x-y)^2 + (y-z)^2 = 1 \), the equation becomes more manageable. Each term now mirrors a perfect square, paving the way to derive plane equations.
Through this transformation, the initial equation's complexity lessens, and newer insights naturally arise.
Equation Reorganization
Reorganizing equations is crucial when dealing with complex mathematical expressions. This process involves rearranging terms, combining like terms, and sometimes factoring to reveal simpler forms within complicated expressions.
Here, the initial equation \( x^2 + y^2 + z^2 - 2xy - 2xz + 2yz - 1 = 0 \) seems daunting at first glance. Reorganization aids in restructuring this equation, allowing us to identify terms that can complete the square, simplifying the equation further and eventually transforming it into recognizable plane equations.
With careful attention to structure and operation, reorganization sets the stage for breaking down complexities.
Planes in 3D Geometry
In 3D geometry, planes are flat, two-dimensional surfaces that extend infinitely in three dimensions. Understanding their equations is vital in analyzing spatial relationships.
The plane equations 漏x - y = 1漏 and 漏y - z = -1漏, derived from the original problem, are standard representations where 漏Ax + By + Cz = D漏 denotes the equation of a plane with normal vector components (A, B, C).
Within the context of this exercise, reducing a complex surface down to simple plane equations showcases the utility of geometric understanding when solving algebraic challenges, linking algebra to spatial visualizations effectively.

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Most popular questions from this chapter

[T] In cartography, Earth is approximated by an oblate spheroid rather than a sphere. The radii at the equator and poles are approximately \(3963 \mathrm{mi}\) and \(3950 \mathrm{mi}\), respectively. a. Write the equation in standard form of the ellipsoid that represents the shape of Earth. Assume the center of Earth is at the origin and that the trace formed by plane \(z=0\) corresponds to the equator. b. Sketch the graph. c. Find the equation of the intersection curve of the surface with plane \(z=1000\) that is parallel to the \(x y\) -plane. The intersection curve is called a parallel. d. Find the equation of the intersection curve of the surface with plane \(x+y=0\) that passes through the \(z\) -axis. The intersection curve is called a meridian.

For the following exercises, the equation of a quadric surface is given. a. Use the method of completing the square to write the equation in standard form. b. Identify the surface. \(x^{2}-y^{2}+z^{2}-12 z+2 x+37=0\)

Use the following figure as an aid in identifying the relationship between the rectangular, cylindrical, and spherical coordinate systems. For the following exercises, the cylindrical coordinates \((r, \theta, z)\) of a point are given. Find the rectangular coordinates \((x, y, z)\) of the point. \((2, \pi,-4)\)

For the following exercises, find the equation of the plane with the given properties. The plane that passes through point (4,7,-1) and has normal vector \(\mathbf{n}=\langle 3,4,2\rangle\)

For the following exercises, the equation of a surface in spherical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface. \([\mathrm{T}] \rho=2 \cos \varphi\)
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