Problem 42 - Advanced Engineering Mathematics

Question

Convert the point given in rectangular cocrdinates to cylindrical coondinates. $$ (1,2,7) $$

Step-by-Step Solution

Answer

The cylindrical coordinates are \((\sqrt{5}, \tan^{-1}(2), 7)\).

Step 1: Understanding Cylindrical Coordinates

Cylindrical coordinates consist of three parts: \((r, \theta, z)\), where \(r\) is the radial distance from the z-axis, \(\theta\) is the angle relative to the positive x-axis, and \(z\) is the height component, which remains the same as in rectangular coordinates.

Step 2: Calculate Radial Distance \(r\)

To find \(r\), use the formula: \[r = \sqrt{x^2 + y^2}\]For the point \((1, 2, 7)\), we have \(x = 1\) and \(y = 2\). Substitute these into the formula to get:\[r = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}\].

Step 3: Calculate the Angle \(\theta\)

To find \(\theta\), use the arctangent function:\[\theta = \tan^{-1}\left(\frac{y}{x}\right)\]Substitute \(x = 1\) and \(y = 2\) to find: \[\theta = \tan^{-1}\left(\frac{2}{1}\right) = \tan^{-1}(2)\].

Step 4: Retain the Height \(z\)

In cylindrical coordinates, the \(z\)-coordinate remains the same. Therefore, \(z = 7\) for the original point \((1, 2, 7)\).

Step 5: Assemble Cylindrical Coordinates

Combine the results from previous steps to express the point in cylindrical coordinates:\[(r, \theta, z) = \left(\sqrt{5}, \tan^{-1}(2), 7\right)\].

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way of representing points in a plane or space using distances along perpendicular axes. In three dimensions, these coordinates are typically denoted as

• \((x, y, z)\)

where:

• \(x\) is the distance along the horizontal (x-axis),
• \(y\) is the distance along the vertical (y-axis), and
• \(z\) is the distance along the depth (z-axis).

These coordinates allow for precise location marking, utilizing perpendicular direction measurements.

In geometry and physics, understanding this coordinate system is crucial, as it serves as a foundational tool for problem-solving and analysis.

Coordinate Conversion

Coordinate conversion between rectangular and cylindrical coordinates involves transforming the representation of a point from one system to another. This process is essential in applications where different coordinate systems provide more effective solutions or interpretations of a problem.To convert from rectangular coordinates

• \((x, y, z)\)

to cylindrical coordinates

• \((r, \theta, z)\)

follow these steps:

• Calculate the radial distance \(r\) using the formula \(r = \sqrt{x^2 + y^2}\).
• Determine the angle \(\theta\) by \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\).
• Retain the \(z\)-coordinate as it remains unchanged.

This conversion is particularly useful in situations where symmetry about the z-axis simplifies analysis, such as in cylindrical structures or around rotational axes.

Radial Distance Calculation

Radial distance is a component of cylindrical coordinates and represents the distance from a point to the z-axis. Calculating this is a straightforward yet crucial step in converting coordinates.To find the radial distance \(r\) from a point \((x, y, z)\) in rectangular coordinates:

• Use the formula \(r = \sqrt{x^2 + y^2}\).

For instance, in the example point \((1, 2, 7)\):

• Substitute \(x = 1\) and \(y = 2\) into the formula.
• This yields \(r = \sqrt{1^2 + 2^2} = \sqrt{5}\).

Understanding radial distance is vital for describing positions relative to the z-axis, aiding in numerous engineering and physics problems.

Angle Calculation in Cylindrical Coordinates

In cylindrical coordinates, the angle \(\theta\) describes the rotation of a point around the z-axis, starting from the positive x-axis. Calculating this angle helps fully define the location of a point in a circular plane.To find \(\theta\) for a point \((x, y)\) in rectangular coordinates:

• Utilize the arctangent function: \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\).

For the example point \((1, 2, 7)\):

• Substitute to get \(\theta = \tan^{-1}\left(\frac{2}{1}\right) = \tan^{-1}(2)\).

This calculation is essential for applications involving rotational dynamics and scenarios where angular positioning is needed, such as in robotic arms and navigation.