Question

Use a computer algebra system to graph a view of the cylinder \(y^{2}+z^{2}=4\) from the points (a) \((10,0,0),\) (b) \((0,10,0),\) and (c) (10,10,10)

Short answer

By plotting the cylinder equation, it could be observed that the cylinder appears differently when viewed from different points in the 3D space. From (10,0,0), the cylinder appears as a circle of radius 2 as it is viewed along the length (parallel to x-axis). From (0,10,0) it would also appear as a circle of radius 2, but now along the y-axis. From (10,10,10), the cylinder appears more oblique, providing a clear view of the circle and its orientation along the x-axis.

Step-by-step solution

Understanding the Figure

The equation \(y^{2}+z^{2}=4\) represents a cylinder of radius 2 in three dimensional space. It is oriented along the x-axis, because x is absent in the equation.

Graphing the Cylinder

Using a computer algebra system or a graphing calculator capable of 3D plotting, the cylinder can graphed. This step is best done using an interactive system to assist with understanding the orientation of the figure in 3D space.

View from point (10,0,0)

In the computer algebra system, adjust the viewpoint to be from the point (10,0,0). This would be a visual representation of the cylinder from along the positive x-axis. Observations should be in relation to the x-axis.

View from point (0,10,0)

Next, adjust the viewpoint in the system to be from (0,10,0). This is a view from the y-axis. Observations should be made about how the figure is seen from this perspective.

View from point (10,10,10)

Finally, adjust the viewpoint to (10,10,10). This is a view from the point in space located at the same positive distance from the origin along all axes. Observe and describe how the cylinder appears from this angle.

Key concepts

3D graphing techniques

Three-dimensional (3D) graphing is a visual representation of objects in a three-dimensional space. This technique is essential in fields such as mathematics, engineering, and the physical sciences to illustrate the relationship between various variables. When graphing a cylinder, as in the equation \(y^{2}+z^{2}=4\), different techniques can be employed to better understand its structure.

Firstly, knowing that the absence of the x-variable indicates that the cylinder is oriented along the x-axis is vital. Once that is clear, using 3D graphing software or tools is the next step. These tools often come with interactive features such as zoom, rotate, and pan that enable users to view the cylinder from various angles and distances, giving a comprehensive understanding of its geometry.

To improve on textbook graphing exercises, it can be helpful to include multiple views of the same object. This highlights the importance of perspective in 3D graphing, as objects can appear very different when viewed from different points. By engaging with the graph interactively, students can better grasp the dimensions and orientation of 3D figures like cylinders.

Computer algebra systems

Computer Algebra Systems (CAS) are software tools designed to simplify and solve algebraic problems. They are particularly useful when dealing with complex mathematical functions and shapes, including those in three dimensions. A CAS can symbolically manipulate mathematical expressions, simulating what would happen if you were to solve the expression by hand.

When graphing cylinders in 3D, a CAS becomes exceptionally relevant. The cylinder from our exercise has the equation \(y^{2}+z^{2}=4\), which can be challenging to visualize without technology. By using a CAS, you can input an equation and have the system generate a visual representation quickly. This tool is invaluable for students because it allows for visual learning and enhances comprehension. They can easily rotate the graph, change viewpoints, and manipulate the figure to observe it from the points \((10,0,0)\), \((0,10,0)\), and \((10,10,10)\), which would support their spatial understanding of the geometry in question.

Cylindrical coordinates

Cylindrical coordinates offer another perspective when dealing with three-dimensional shapes and are especially adept for handling objects like cylinders that have natural symmetry around one axis. This coordinate system is defined by a radius \(r\), an angle \(\theta\), and a height \(z\), directly corresponding to the intuitively cylindrical nature of shapes like our example with the equation \(y^{2}+z^{2}=4\).

In this system, you can describe the location of a point in space relative to a cylindrical structure, rather than with the usual x, y, and z coordinates. It's useful for graphing cylindrical shapes because it aligns with the cylinder's natural geometry, simplifying the understanding and calculations of distances and angles around the cylinder. By converting between rectangular and cylindrical coordinates, students can gain an improved perspective on how the cylindrical shape occupies space and how it can be represented from different viewpoints, which reinforces comprehension of both the coordinate system and the 3D structure of the cylinder.