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Chapter 12: Problem 13

In Exercises \(1-16,\) give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$x^{2}+y^{2}=4, \quad z=y$$

Short Answer

Expert verified
The set of points is a helical curve on a cylinder of radius 2 with axis along the \(z\)-axis, satisfying \(z = y\).

Step by step solution

01

Analyze the first equation

The first equation is \(x^2 + y^2 = 4\). This is the equation of a circle in the \(xy\)-plane, with its center at the origin \((0, 0)\) and a radius of 2. Therefore, in 3D space, this represents a cylinder extending infinitely along the \(z\)-axis.
02

Analyze the second equation

The second equation is \(z = y\). This equation implies that for any point in the space, the \(z\)-coordinate is always equal to the \(y\)-coordinate. This describes a plane in 3D space that is tilted, forming a 45-degree angle with the \(y\)-axis and the \(z\)-axis.
03

Combine the constraints

To find the set of points satisfying both equations, we consider the intersection of the geometric figures described. The cylinder defined by \(x^2 + y^2 = 4\) intersects the plane \(z = y\). The intersection results in a curve where every point on the cylinder has its height (\(z\)) equal to its \(y\)-coordinate. Visually, this means slicing the cylinder with the plane to see the elliptical cross-section.
04

Visualize the intersection

To visualize the intersection, consider traces of the cylinder. For each point \((x, y)\) on the circular base where \(x^2 + y^2 = 4\), assign \(z = y\). This yields a helical curve wrapping around the cylinder because the \(z\)-value follows the height attributed by its corresponding \(y\)-value on the circle.
05

Provide the geometric description

The geometric description of the set of points is a helical curve inscribed on a cylindrical surface with radius 2, where the cylinder's axis is along the line \(x = 0, \; y = 0, \; z\) and every point satisfies \(z = y\). The helix spirals upwards (or downwards) along the surface, creating a tilted spiral pattern by the nature of \(z = y\).

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Key Concepts

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

Cylinder Equation
The equation given by \(x^2 + y^2 = 4\) defines a cylinder in three-dimensional space. On the 2D plane, this equation represents a circle centered at the origin \((0, 0)\) with a radius of 2. However, when considering three-dimensional geometry, this becomes a circular cylinder extending infinitely along the \(z\)-axis.

Key characteristics of the cylinder:
  • Center: Origin \((0, 0)\) in the \(xy\)-plane.
  • Radius: 2.
  • Infinite extension: Along the \(z\)-axis, making it a cylinder, not a circle.
In essence, you can visualize this as a circular shape drawn repeatedly at every point on the \(z\)-axis, creating a cylindrical shell. Each circle鈥檚 radius is consistent as you move up and down the \(z\)-axis.
Plane Equation
The equation \(z = y\) defines a plane in three-dimensional space. Here, each point on the plane satisfies the condition that the \(z\)-coordinate is equal to the \(y\)-coordinate.

This plane is slightly tilted, forming a 45-degree angle between both the \(y\)-axis and \(z\)-axis.
  • The angle arises because changes in \(y\) directly cause exact, equal changes in \(z\). Hence, a slope of 1 on the \(y\)-\(z\) plane.
By understanding where this plane lies, you can determine its role in cutting through the cylinder. Picture the plane like a sheet that slices through space maintaining its tilt, indicating the relationship between \(z\) and \(y\).
Helical Curve Intersection
As the plane \(z = y\) intersects with the cylindrical surface described by \(x^2 + y^2 = 4\), a distinct type of curve forms: a helical curve.

The notion of a helix often brings to mind the shape of a spring or a spiral staircase. In this case, the helical curve wraps around the cylinder. This happens because the height \(z\) of any point on the cylinder is dictated by its \(y\)-coordinate, which spirals along as you trace the cylinder.

Think of it like this:
  • For every position \((x, y)\) on the circle that forms the base of the cylinder, assign a corresponding \(z\) equal to \(y\).
  • This creates a continuous incline around the cylinder.
Through this understanding, this helical structure can be visualized as wrapping around the cylindrical surface, making its path seem like a spiraling ascent or descent.
Three-Dimensional Geometry
Understanding three-dimensional geometry involves visualizing components like points, lines, planes, and shapes in space. Here, both the cylinder and the plane interact within this three-dimensional setting.

Always remember the following concepts:
  • Planes: Infinite in two directions, they can slice through three-dimensional objects, creating intersection lines or curves.
  • Cylinders: Such shapes are extended in one direction (in this case, \(z\)-axis), but maintain a consistent circular cross-section.
  • Intersections: These define where two geometric figures meet, forming new shapes or paths, like the helical curve.
In essence, three-dimensional geometry allows us to break down complex forms into their fundamental elements, thereby understanding the resulting intersections and shapes. It's a matter of linking flat representations to how they manifest in 3D space.

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Most popular questions from this chapter

Using the definition of the projection of \(\mathbf{u}\) onto \(\mathbf{v},\) show by direct calculation that \(\left(\mathbf{u}-\operatorname{proj}_{\mathbf{v}} \mathbf{u}\right) \cdot \operatorname{proj}_{\mathbf{v}} \mathbf{u}=0\)

In Exercises \(1-8,\) find the length and direction (when defined) of \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u} .\) $$ \mathbf{u}=2 \mathbf{i}+3 \mathbf{j}, \quad \mathbf{v}=-\mathbf{i}+\mathbf{j} $$

Cross products of three vectors Show that except in degenerate cases, \((\mathbf{u} \times \mathbf{v}) \times \mathbf{w}\) lies in the plane of \(\mathbf{u}\) and \(\mathbf{v},\) whereas \(\mathbf{u} \times(\mathbf{v} \times \mathbf{w})\) lies in the plane of \(\mathbf{v}\) and \(\mathbf{w} .\) What are the degenerate cases?

In Exercises \(9-14,\) sketch the coordinate axes and then include the vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{u} \times \mathbf{v}\) as vectors starting at the origin. $$ \mathbf{u}=\mathbf{i}+\mathbf{j}, \quad \mathbf{v}=\mathbf{i}-\mathbf{j} $$

Dot multiplication is positive definite Show that dot multiplication of vectors is positive definite; that is, show that \(\mathbf{u} \cdot \mathbf{u} \geq 0\) for every vector \(\mathbf{u}\) and that \(\mathbf{u} \cdot \mathbf{u}=0\) if and only if \(\mathbf{u}=\mathbf{0}\) .
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