Problem 42 - Advanced Engineering Mathematics

Question

In Problems \(39-42\), convert the point given in rectangular coardinates to cylindrical coordinates. $$ (1,2,7) $$

Step-by-Step Solution

Answer

The point in cylindrical coordinates is \((\sqrt{5}, \arctan(2), 7)\).

Step 1: Identify Components

The point assigned in rectangular coordinates is given as \((x, y, z) = (1, 2, 7)\). In cylindrical coordinates, the point is expressed as \((r, \theta, z)\). The third coordinate, \(z\), remains the same as in rectangular coordinates.

Step 2: Calculate r (Radial Distance)

The formula to calculate \(r\) (the radial distance from the z-axis) is \(r = \sqrt{x^2 + y^2}\). Use this formula with \(x = 1\) and \(y = 2\): \[r = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}\]. So, \(r = \sqrt{5}\).

Step 3: Calculate θ (Angular Component)

Determine \(\theta\), the angle in radians from the positive x-axis to the projection of the point in the xy-plane, using the formula \(\theta = \arctan\left(\frac{y}{x}\right)\). Plug in \(x = 1\) and \(y = 2\): \[\theta = \arctan\left(\frac{2}{1}\right) = \arctan(2)\]. This is the value for \(\theta\).

Step 4: Present the Cylindrical Coordinates

The cylindrical coordinates are presented as \((r, \theta, z)\). From the previous steps, \(r = \sqrt{5}\), \(\theta = \arctan(2)\), and \(z = 7\). Therefore, the point in cylindrical coordinates is \((\sqrt{5}, \arctan(2), 7)\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way to specify the position of a point in space using three values:

• the x-coordinate, representing the horizontal distance along the x-axis,
• the y-coordinate, representing the vertical distance along the y-axis,
• the z-coordinate, which denotes the depth or height above the xy-plane.

In our exercise, the point (1, 2, 7) was given in this format. This means the point is 1 unit along the x-axis, 2 units along the y-axis, and 7 units above the xy-plane. This coordinate system is quite useful for straightforward calculations and is widely used in various fields like graphics, physics, and engineering. Consequently, converting these coordinates into cylindrical coordinates helps solve problems where rotational symmetry around an axis is involved.

Coordinate Conversion

Coordinate conversion is the process of switching from one coordinate system to another, in this case from rectangular to cylindrical coordinates. This conversion is crucial in fields that involve rotational or circular motion, such as fluid dynamics or electromagnetism. Our goal was to convert the point (1, 2, 7) from rectangular to cylindrical coordinates.

• Firstly, we keep the z-coordinate the same.
• We then calculate the radial distance (r) and angular component (θ) to express the point in cylindrical coordinates.

Both are derived from the x and y rectangular coordinates. This method allows us to better understand how a point relates to a particular axis (the z-axis, in this case), making it easier to solve specific types of geometric problems.

Radial Distance

Radial distance, noted as \(r\), measures how far a point is from the z-axis. It's one of the key parameters in cylindrical coordinates. We use the formula \(r = \sqrt{x^2 + y^2}\) to find this distance using the x and y components of a given point.
For the point (1, 2, 7), we applied this formula as follows:
\[r = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}\]Hence, the radial distance \(r\) is \(\sqrt{5}\). Understanding \(r\) is fundamental in determining how a point appears when projected onto the z-plane. It plays a significant role in simplifying the analysis of systems with symmetrical structures, as it represents the exact circular radius from the central axis.

Angular Component

The angular component \(\theta\) in cylindrical coordinates represents the angle measured in radians from the positive x-axis towards the projection of the vector onto the xy-plane. It is a critical component to fully describe the position of a point in a rotational context.To compute \(\theta\), we use the formula \(\theta = \arctan\left(\frac{y}{x}\right)\). For the point (1, 2, 7):
\[\theta = \arctan\left(\frac{2}{1}\right) = \arctan(2)\]This means \(\theta\) is the angle that describes how the point lies in relation to the x-axis in the xy-plane. Understanding this component is beneficial in various technical fields as it assists in analyzing and solving problems involving circular movements or patterns within a plane.