Problem 87 - Thomas Calculus

Question

Vertical planes in cylindrical coordinates a. Show that planes perpendicular to the \(x\) -axis have equations of the form \(r=a \sec \theta\) in cylindrical coordinates. b. Show that planes perpendicular to the \(y\) -axis have equations of the form \(r=b \csc \theta\)

Step-by-Step Solution

Answer

a. Planes: \(r = a \sec \theta\); b. Planes: \(r = b \csc \theta\).

Step 1: Understand the Problem

In this exercise, we need to convert Cartesian equations of planes in the form perpendicular to the x-axis and y-axis into cylindrical coordinates, where polar coordinates are given by \( (r, \theta, z) \).

Step 2: Convert Equation to Cylindrical Coordinates for Part (a)

For planes perpendicular to the \(x\)-axis, the equation in Cartesian form is \(x = a\). Using the equations \(x = r \cos \theta\) and \(z = z\) with the definitions of cylindrical coordinates, we need to adjust this equation accordingly.

Step 3: Convert x = a to Cylindrical Coordinates

Replace \(x\) in the equation \(x = a\) with \(r \cos \theta\). This gives us the equation \(r \cos \theta = a\).

Step 4: Solve for r in terms of θ for Part (a)

To solve \(r \cos \theta = a\) for \(r\), we rearrange the equation as \(r = \frac{a}{\cos \theta}\), which is equivalent to \(r = a \sec \theta\). This confirms the required form for planes perpendicular to the \(x\)-axis in cylindrical coordinates.

Step 5: Convert Equation to Cylindrical Coordinates for Part (b)

For planes perpendicular to the \(y\)-axis, the equation in Cartesian form is \(y = b\). Using the equations \(y = r \sin \theta\), \(x = r \cos \theta\) and \(z = z\), we transform this equation into cylindrical coordinates.

Step 6: Convert y = b to Cylindrical Coordinates

Replace \(y\) in the equation \(y = b\) with \(r \sin \theta\). This results in the equation \(r \sin \theta = b\).

Step 7: Solve for r in terms of θ for Part (b)

To get \(r\) in terms of \(\theta\), rewrite \(r \sin \theta = b\) as \(r = \frac{b}{\sin \theta}\). This simplifies to \(r = b \csc \theta\), confirming the required form for planes perpendicular to the \(y\)-axis.

Key Concepts

Cartesian to Cylindrical Conversion

Converting from Cartesian coordinates to cylindrical coordinates can be quite straightforward when you understand the basics. Imagine you have a point in a three-dimensional Cartesian system, represented as \( (x, y, z) \). Cylindrical coordinates bring an interesting twist by introducing polar coordinates into the mix, encapsulated as \( (r, \theta, z) \). To perform the conversion, there's a simple set of relationships:

• \( r = \sqrt{x^2 + y^2} \) — This is the radial distance from the z-axis.
• \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \) — This angle locates the point in the polar plane.
• \( z = z \) — This remains unchanged.

These conversions help transform 3D points so that they can be analyzed in terms of both direction and height. Hence, understanding these relationships allows you to easily switch and derive equations in cylindrical coordinates, as shown in the exercise above for planes perpendicular to the x and y axes.

Polar Coordinates

Polar coordinates form a substantial part of cylindrical coordinates and deal primarily with locations in a plane. Specifically, each point is identified by \( (r, \theta) \), where:

• \( r \) is the radial distance from the origin to the point.
• \( \theta \) is the angular displacement measured from the positive x-axis, moving counterclockwise.

These coordinates are especially useful for dealing with curves and shapes that have rotational symmetry. For instance, when converting the plane equations from Cartesian to cylindrical, polar coordinates allow the expression of \( x = a \) or \( y = b \) through trigonometric identities involving \( \cos \theta \) and \( \sin \theta \) respectively. This not only presents a new form of describing lines and curves but also simplifies mathematical manipulations when symmetry is present.

Planes in Coordinate Geometry

Planes in coordinate geometry can be a bit tricky but boil down to simple forms with consistent rules. When planes are described in Cartesian coordinates, they generally take the shape \( Ax + By + Cz = D \). However, they can manifest as simpler forms when aligned with coordinate axes, like \( x = a \) or \( y = b \) which were seen in the exercise.In terms of geometry:

• A plane perpendicular to the x-axis has an equation involving only \( x \) or \( r \, \cos \theta \) in cylindrical coordinates.
• If a plane is perpendicular to the y-axis, it involves \( y \) or \( r \, \sin \theta \) in its equation form.

What cylindrical coordinates offer in such scenarios is a new lens to better visualize and solve problems involving planes, especially when they intersect in various ways relative to the axes. The equations derived in cylindrical forms, like \( r = a \sec \theta \), are a testament to their utility in simplifying complex spatial equations.