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Chapter 2: Problem 329

For the following exercises, rewrite the given equation of the quadric surface in standard form. Identify the surface. $$ 6 x=3 y^{2}+2 z^{2} $$

Short Answer

Expert verified
The surface is an elliptical paraboloid.

Step by step solution

01

Identify the Quadric Surface Equation

The given equation of the quadric surface is \( 6x = 3y^2 + 2z^2 \). This equation contains three terms: a linear term in \( x \), and quadratic terms in \( y \) and \( z \). Our task is to rewrite this equation in standard form.
02

Divide by Constants to Simplify

First, divide the entire equation by 6 to make the equation easier to handle and to see clear coefficients for each variable: \[ x = \frac{1}{2}y^2 + \frac{1}{3}z^2. \] This division gives us clearer coefficients, allowing us to better recognize the surface form.
03

Recognize the Standard Form

Compare the simplified equation \( x = \frac{1}{2}y^2 + \frac{1}{3}z^2 \) to known standard forms of quadric surfaces. The form \( x = Ay^2 + Bz^2 \) shows that this equation resembles the standard form of an elliptical paraboloid, where \( x \) is the linear term and \( y^2, z^2 \) are quadratic terms.
04

Identify the Surface

The final standard form \( x = \frac{1}{2}y^2 + \frac{1}{3}z^2 \) fits the description of an elliptical paraboloid. In this surface, as \( y \) and \( z \) vary, \( x \) is defined by the quadratic combinations of these two variables, creating a bowl-shaped surface.

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Key Concepts

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

Elliptical Paraboloid
An elliptical paraboloid is a fascinating shape that appears often in mathematics and physics. It resembles a three-dimensional bowl or a dome, curving upwards or downwards depending on the signs of its quadratic terms. This shape is a type of quadric surface defined by an equation where one variable is expressed linearly, and the other two variables are squared.

Typical equations might look like: \( z = Ax^2 + By^2 \) or \( x = Ay^2 + Bz^2 \), where A and B are constants. Here, the linear term (such as \( z \) or \( x \)) denotes the axis of symmetry, which is aligned with the direction the paraboloid opens.
  • Openings: The paraboloid opens upwards if both constants A and B are positive.
  • Shapes: You get a round bowl if A equals B, resulting in a circular cross-section. If A differs from B, the cross-section is elliptical, appearing elongated along one axis.
Elliptical paraboloids have practical applications in architecture and engineering, such as in satellite dishes and parabolic reflectors.
Quadric Surface Equation
A quadric surface is represented by a polynomial equation of degree two in three-dimensional space. These surfaces are vital for understanding the geometric shapes that appear in various scientific fields.

The equation \( Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0 \) encapsulates a wide family of surfaces. Each equation's specific combination of terms (linear, quadratic, or mixed) determines the kind of surface formed.
  • Types: Common types include ellipsoids, hyperboloids, paraboloids, and more.
  • Coefficients: The values and signs of the coefficients (A, B, C...) greatly influence the surface's shape and orientation.
By analyzing these equations, mathematicians can predict the behavior of physical systems or design the architecture for complex structures.
Standard Form of Quadratic Surfaces
Converting a quadric surface equation into its standard form can illuminate its geometric characteristics. The standard form refines the equation by eliminating variable mix-ups and simplifying its representation, making it easier to identify the surface type.

The standard form often emphasizes symmetry and structural clarity. For instance, transforming \( 6x = 3y^2 + 2z^2 \) into \( x = \frac{1}{2}y^2 + \frac{1}{3}z^2 \) helps recognize the surface as an elliptical paraboloid.
  • Procedure: It typically involves dividing by a constant to simplify coefficients, as in the example, leading to a clearer viewpoint.
  • Comparison: Matching the equation鈥檚 form with known standard types (ellipsoid, hyperboloid, paraboloid) facilitates easy identification.
Standard forms are crucial in solving geometric problems and understanding spatial dimensions in multiple fields.

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Most popular questions from this chapter

For the following exercises, the equation of a plane is given. a. Find normal vector \(\mathbf{n}\) to the plane. Express \(\mathbf{n}\) using standard unit vectors. b. Find the intersections of the plane with the axes of coordinates. c. Sketch the plane. \(3 x+4 y-12=0\)

For the following exercises, the equations of two planes are given. a. Determine whether the planes are parallel, orthogonal, or neither. b. If the planes are neither parallel nor orthogonal, then find the measure of the angle between the planes. Express the answer in degrees rounded to the nearest integer. \(x-5 y-z=1, \quad 5 x-25 y-5 z=-3\)

For the following exercises, find the unit vectors. Find the unit vector that has the same direction as vector \(\mathbf{v}\) that begins at \((1,4,10)\) and ends at \((3,0,4) .\)

For the following exercises, the equation of a plane is given. a. Find normal vector \(\mathbf{n}\) to the plane. Express \(\mathbf{n}\) using standard unit vectors. b. Find the intersections of the plane with the axes of coordinates. c. Sketch the plane. \(x+z=0\)

For the following exercises, consider a small boat crossing a river. If the boat velocity is 5 \(\mathrm{km} / \mathrm{h}\) due north in still water and the water has a current of 2 \(\mathrm{km} / \mathrm{h}\) due west (see the following figure), what is the velocity of the boat relative to shore? What is the angle \(\theta\) that the boat is actually traveling?
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