Problem 37 - Advanced Engineering Mathematics

Question

Convert the point given in cylindrical cocudinates to rectangular cocrdinates. $$ \left(\sqrt{3}, \frac{\pi}{3},-4\right) $$

Step-by-Step Solution

Answer

The point in rectangular coordinates is \(\left(\frac{\sqrt{3}}{2}, \frac{3}{2}, -4\right)\).

Step 1: Understand Cylindrical Coordinates

Cylindrical coordinates are represented as \((r, \theta, z)\) where \(r\) is the radial distance from the origin in the xy-plane, \(\theta\) is the angle measured counterclockwise from the positive x-axis, and \(z\) is the height above the xy-plane. The given point is \((\sqrt{3}, \frac{\pi}{3}, -4)\).

Step 2: Convert the Radial and Angular Components

To convert the radial and angular components \((r, \theta)\) from cylindrical to rectangular coordinates, we use the formulas: \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). By substituting the values given, \(x = \sqrt{3} \cos\left(\frac{\pi}{3}\right)\) and \(y = \sqrt{3} \sin\left(\frac{\pi}{3}\right)\).

Step 3: Calculate x Component

Calculate \(x\) using the formula \(x = r \cos(\theta)\): \[x = \sqrt{3} \cos\left(\frac{\pi}{3}\right) = \sqrt{3} \times \frac{1}{2} = \frac{\sqrt{3}}{2}\].

Step 4: Calculate y Component

Calculate \(y\) using the formula \(y = r \sin(\theta)\): \[y = \sqrt{3} \sin\left(\frac{\pi}{3}\right) = \sqrt{3} \times \frac{\sqrt{3}}{2} = \frac{3}{2}\].

Step 5: Assign z Component

In rectangular coordinates, the \(z\)-component remains the same as in cylindrical coordinates. So, \(z = -4\).

Step 6: Compile the Rectangular Coordinates

Now we combine all calculated components to form the rectangular coordinates: \(\left(\frac{\sqrt{3}}{2}, \frac{3}{2}, -4\right)\).

Key Concepts

Cylindrical Coordinates

Cylindrical coordinates are a three-dimensional coordinate system that extends polar coordinates by including height as a dimension.
Simply put, it consists of three components:

• Radial Distance \( (r) \): This is the distance from a point to the origin within the xy-plane.
• Angular Component \( (\theta) \): This angle measures how far you rotate from the positive x-axis.
• Height \( (z) \): The vertical distance above the xy-plane, which remains unchanged when converting to rectangular coordinates.

This system is particularly useful when dealing with problems lacking clear symmetry in rectangular coordinates, or when seamless transitions between vertical and radial distances are advantageous.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are the most familiar system used in mathematics.
They are represented by the triplet \((x, y, z)\). Here’s how each part functions:

• \( x \): Measures how far a point is along the horizontal axis.
• \( y \): Measures how far a point is along the vertical axis in the xy-plane.
• \( z \): Represents the height, similar to in cylindrical coordinates.

Rectangular coordinates are commonly used in many engineering and physics problems due to their straightforward nature, providing a clear grid system for x, y, and z directions that simplifies mathematical manipulation and analysis.

Mathematical Transformation

Mathematical transformation involves converting one set of coordinates to another.
In this case, transforming from cylindrical to rectangular coordinates involves straightforward trigonometric calculations:

• For the x-component, use \( x = r \cos(\theta) \).
• For the y-component, use \( y = r \sin(\theta) \).

These transformations are pivotal in mathematics, especially when solving problems involving geometry and physics where different viewpoints or perspectives enhance comprehension or calculation efficiency.

Engineering Mathematics

In engineering mathematics, coordinate systems are regularly employed to model and solve real-world engineering problems.
Cylindrical and rectangular coordinates each offer unique advantages, depending on the problem:

• Cylindrical coordinates are ideal for modeling scenarios involving rotation or circular symmetry, such as modeling cylindrical pipes or electrical coils.
• Rectangular coordinates are used for analysis in scenarios that conform to perpendicular, or linear boundaries, such as in structural engineering frameworks.

Converting between these systems accurately and efficiently allows engineers to approach and solve complex problems more effectively, optimizing the design and function of systems they create.