Problem 41 - Advanced Engineering Mathematics

Question

Convert the point given in rectangular cocrdinates to cylindrical coondinates. $$ (-\sqrt{2}, \sqrt{6}, 2) $$

Step-by-Step Solution

Answer

The cylindrical coordinates are \((2\sqrt{2}, \frac{2\pi}{3}, 2)\).

Step 1: Identify the Rectangular Coordinates

The point given in rectangular coordinates is \((x, y, z) = (-\sqrt{2}, \sqrt{6}, 2)\).

Step 2: Calculate the Radial Distance

To find the radial distance \(r\) for cylindrical coordinates, use the formula:\[r = \sqrt{x^2 + y^2}\].Substitute the values:\(r = \sqrt{(-\sqrt{2})^2 + (\sqrt{6})^2} = \sqrt{2 + 6} = \sqrt{8} = 2\sqrt{2}\).

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

Calculate the angle \(\theta\) in the xy-plane with the formula:\[\theta = \tan^{-1}\left(\frac{y}{x}\right)\].For our given coordinates:\(\theta = \tan^{-1}\left(\frac{\sqrt{6}}{-\sqrt{2}}\right) = \tan^{-1}\left(-\sqrt{3}\right)\).The angle depends on the quadrant; since \(x < 0\) and \(y > 0\), the point is in the second quadrant. Thus, \(\theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3}\).

Step 4: Determine the z-component

The \(z\)-coordinate remains unchanged in cylindrical coordinates. Therefore, \(z = 2\).

Step 5: Assemble the Cylindrical Coordinates

Combine \(r\), \(\theta\), and \(z\) to express the point in cylindrical coordinates:\((r, \theta, z) = (2\sqrt{2}, \frac{2\pi}{3}, 2)\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates provide a way of pinpointing a location in a 3-dimensional space using three values:

• The x-coordinate which represents the horizontal position towards the east or west.
• The y-coordinate which specifies the vertical position towards the north or south.
• The z-coordinate which gives the height position upwards or downwards.

In this coordinate system, a point is defined as \((x, y, z)\).Given the original task, the rectangular coordinates are \((-\sqrt{2}, \sqrt{6}, 2)\).This notation is intuitive when graphing or visualizing points within a regular grid structure, as it creates a simple and unified view of space.

Radial Distance

In cylindrical coordinates, radial distance measures how far a point is from the z-axis, which acts as the centerline. This distance is denoted by the letter \(r\).The radial distance is found using the Pythagorean theorem: \[r = \sqrt{x^2 + y^2}\].This formula works because, on a flat plane at any height (constant \(z\)), the position is akin to describing a right triangle where \(x\) and \(y\) are the perpendicular sides. In this exercise:

• The x-value is \(-\sqrt{2}\)
• The y-value is \(\sqrt{6}\)
• Using these values, the radial distance is calculated to be \(2\sqrt{2}\).

Radial distance essentially compresses the 2D aspect of the rectangular system into a single measure of distance from a central axis.

Angle Calculation

In cylindrical coordinates, the angle \(\theta\) is used to specify the direction in the xy-plane from the positive x-axis.This angle is crucial for determining the rotational position around the z-axis. For finding the angle:\[\theta = \tan^{-1}\left(\frac{y}{x}\right)\]However, it’s important to consider the quadrant where \(x\) and \(y\) lie to accurately determine the angle.For this example:

• \(x = -\sqrt{2}\) is negative, and
• \(y = \sqrt{6}\) is positive, placing the point in the second quadrant.

In the second quadrant, you must adjust the angle appropriately:\[\theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3}\].This adjustment ensures that \(\theta\) correctly reflects the point's position relative to the axes.

Coordinate Conversion

Converting from rectangular to cylindrical coordinates involves transitioning from the \((x, y, z)\) format into \((r, \theta, z)\). Each component plays a specific role:

• \(r\) represents the radial distance, or how far the point is from the z-axis, calculated through the horizontal (x-y plane) components.
• \(\theta\) gives the angle from the positive x-axis, crucial for direction in the xy plane.
• The z-component remains unchanged, ensuring vertical alignment with the original point.

As a practical illustration of these conversions:

• The example calculation produces the cylindrical coordinates \((2\sqrt{2}, \frac{2\pi}{3}, 2)\), indicating that the point is still at the same height (z) while rotating and shifting its radial position.

These transformations help in scenarios requiring rotational symmetry, such as in physics and engineering applications.