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Chapter 11: Problem 40

Describe the given set with a single equation or with a pair of equations. The circle of radius 1 centered at (-3,4,1) and lying in a plane parallel to the a. \(x y\) -plane b. \(y z\) -plane c. \(x z\) -plane

Short Answer

Expert verified
a. \((x + 3)^2 + (y - 4)^2 = 1\), \(z = 1\); b. \((y - 4)^2 + (z - 1)^2 = 1\), \(x = -3\); c. \((x + 3)^2 + (z - 1)^2 = 1\), \(y = 4\).

Step by step solution

01

Understand the Circle in 3D Space

A circle in 3D space can be defined with a center point and a radius, but its orientation in space is defined based on the plane it lies in. The general form for the equation of a circle in 3D is dependent on its plane of orientation.
02

Define the Circle in the XY-plane

For the circle to lie in a plane parallel to the \(xy\)-plane, the \(z\) coordinate remains constant. The equation would be of the form: \((x + 3)^2 + (y - 4)^2 = 1\), with \(z = 1\).
03

Define the Circle in the YZ-plane

For the circle to lie in a plane parallel to the \(yz\)-plane, the \(x\) coordinate remains constant. The equation would be: \((y - 4)^2 + (z - 1)^2 = 1\), with \(x = -3\).
04

Define the Circle in the XZ-plane

To have the circle in a plane parallel to the \(xz\)-plane, the \(y\) coordinate remains constant. The equation would be: \((x + 3)^2 + (z - 1)^2 = 1\), with \(y = 4\).

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Key Concepts

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

Circle Equations
In 3D geometry, the equation of a circle varies depending on the plane in which it lies. A circle is defined by a center point and a radius. The circle's equation in 3D space highlights its specific orientation through the coordinate that remains fixed across its plane.
  • The equation for a circle in two-dimensional space, such as the coordinate planes ( xy-plane, yz-plane, or xz-plane), is modified by a fixed coordinate representing the circle's perpendicular distance from that plane.
  • For a circle lying in a plane parallel to the xy-plane, the z-coordinate remains constant.
  • Likewise, for circles in a yz-plane or xz-plane, the x-coordinate and y-coordinate remain constant, respectively.
This fixed coordinate directly depends on the offset of the circle from the origin of the coordinate system. Understanding these fundamental differences in circle equations ensures proper representation and orientation of circles within a 3D space.
Planes in 3D Space
Planes in 3D space are crucial in defining the orientation and placement of geometric shapes such as circles. A plane can be mathematically described using an equation of the form \( ax + by + cz = d \). However, in the context of a circle's placement, planes are identified based on which coordinate is perpendicular to the plane.
  • The xy-plane is where the z-coordinate is consistent, meaning a plane parallel to the xy-plane will have a fixed z-value for any point on it.
  • The yz-plane keeps the x-coordinate constant to depict planes extending parallel in 3D space.
  • Similarly, the xz-plane holds the y-value constant, allowing shapes to span across its surface.
This comprehension of planes allows us to translate between 3D positions and their corresponding plane-based dimensions.
Coordinate Geometry
Coordinate geometry, integral to 3D space, uses coordinates to explore and solve geometric problems. It involves analyzing spatial relationships by designating positions using ordered sets of numbers, commonly referred to as coordinates.
  • In a 3D coordinate system, each point is represented by three numbers \((x, y, z)\), each corresponding to a position along the x, y, and z axes.
  • This system allows for precise location mapping of points, shapes, and complexes within three-dimensional environments.
Utilizing coordinate geometry, we can understand and describe complex geometric figures with equations that blend algebra with geometric coordinates. This approach is pivotal in identifying the correct orientation and position of circles in the given planes.
Orientation of Circles
The orientation of circles in 3D space is determined by the plane they are aligned with. This orientation not only affects the circle's equation but also how it interacts spatially within a 3D environment.
  • Circles parallel to the xy-plane maintain their position in the z-axis, with all points having a constant z-coordinate.
  • Those in the yz-plane hold a steady x-coordinate, allowing their size to manifest along the y and z axes.
  • Meanwhile, circles in the xz-plane keep the y-coordinate unchanged, stretching over the x and z dimensions.
Recognizing the orientation of circles is critical, as it dictates how external forces or intersecting shapes might affect their position. It's the basis for many applications in design, modeling, and computational geometry.

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Most popular questions from this chapter

Sketch the surfaces PARABOLOIDS AND CONES $$y=1-x^{2}-z^{2}$$

Sketch the surfaces ASSORTED $$x^{2}+y^{2}=z$$

Unit vectors in the plane \(\quad\) Show that a unit vector in the plane can be expressed as \(\mathbf{u}=(\cos \theta) \mathbf{i}+(\sin \theta) \mathbf{j},\) obtained by rotating \(i\) through an angle \(\theta\) in the counterclockwise direction. Explain why this form gives every unit vector in the plane.

Determine whether the given points are coplanar. $$A(0,0,4), \quad B(6,2,0), \quad C(2,-1,1), \quad D(-3,-4,3)$$

Sketch the surfaces HYPERBOLIC PARABOLOIDS $$x^{2}-y^{2}=z$$
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