Problem 38 - Advanced Engineering Mathematics

Question

Convert the point given in cylindrical cocudinates to rectangular cocrdinates. $$ \left(4, \frac{7 \pi}{4}, 0\right) $$

Step-by-Step Solution

Answer

The point in rectangular coordinates is \((2\sqrt{2}, -2\sqrt{2}, 0)\).

Step 1: Understanding the Provided Point

The given point in cylindrical coordinates is \((r, \theta, z) = (4, \frac{7\pi}{4}, 0)\). Here, \(r\) is the radial distance from the origin, \(\theta\) is the angle in the xy-plane from the positive x-axis, and \(z\) is the height along the z-axis.

Step 2: Applying Conversion Formulas

To convert cylindrical coordinates \((r, \theta, z)\) to rectangular coordinates \((x, y, z)\), we use the formulas: \[x = r \cos \theta,\]\[y = r \sin \theta,\]\[z = z.\]

Step 3: Substituting Values for x

Substitute \(r = 4\) and \(\theta = \frac{7\pi}{4}\) into the formula for \(x\):\[x = 4 \cos \left(\frac{7\pi}{4}\right).\]

Step 4: Calculating x

Calculate the cosine of \(\theta = \frac{7\pi}{4}\):\[\cos \left(\frac{7\pi}{4}\right) = \cos \left(\frac{8\pi}{4} - \frac{\pi}{4}\right) = \cos \left(2\pi - \frac{\pi}{4}\right) = \cos \left(-\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}.\]Therefore, \[x = 4 \times \frac{1}{\sqrt{2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2}.\]

Step 5: Substituting Values for y

Substitute \(r = 4\) and \(\theta = \frac{7\pi}{4}\) into the formula for \(y\):\[y = 4 \sin \left(\frac{7\pi}{4}\right).\]

Step 6: Calculating y

Calculate the sine of \(\theta = \frac{7\pi}{4}\):\[\sin \left(\frac{7\pi}{4}\right) = \sin \left(\frac{8\pi}{4} - \frac{\pi}{4}\right) = \sin \left(2\pi - \frac{\pi}{4}\right) = \sin \left(-\frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}}.\]Therefore, \[y = 4 \times -\frac{1}{\sqrt{2}} = -\frac{4}{\sqrt{2}} = -2\sqrt{2}.\]

Step 7: Using the z-coordinate

The z-coordinate in cylindrical and rectangular coordinates is the same. Therefore, \[z = 0.\]

Step 8: Final Rectangular Coordinates

Combining the results, the rectangular coordinates are: \((x, y, z) = (2\sqrt{2}, -2\sqrt{2}, 0)\).

Key Concepts

Cylindrical Coordinates

Cylindrical coordinates are a system used to define the position of a point in a three-dimensional space using three values. This system is somewhat like polar coordinates but with an added height measurement. It consists of:

• Radial distance \(r\): The distance from the origin (center of the coordinate system) to the point in the xy-plane.
• Angular coordinate \(\theta\): The angle measured from the positive x-axis to the line connecting the origin with the projection of the point in the xy-plane.
• Height \(z\): The vertical distance from the xy-plane to the point.

Cylindrical coordinates are expressed as \((r, \theta, z)\), making them particularly useful for objects and systems with rotational symmetry, such as cylinders or turbines. These coordinates simplify calculations by leveraging the repeated patterns and curves of circular shapes.

Rectangular Coordinates

Rectangular, or Cartesian coordinates, describe the position of points in space using orthogonal axes. These coordinates rely on three coordinates:

• x-coordinate: This is the horizontal distance from the yz-plane to the point.
• y-coordinate: The horizontal distance from the xz-plane to the point.
• z-coordinate: The vertical distance from the xy-plane to the point.

The coordinates \((x, y, z)\) are typical for representing points in most everyday contexts and mathematical computations. Points and distances are measured in straight lines, matched to their respective axes, providing a straightforward way to express locations and geometric entities.

Trigonometric Functions

Trigonometric functions are essential tools for connecting angles with side ratios in right triangles and are used in coordinate conversions. The primary trigonometric functions that relate to coordinate conversions are:

• Cosine \(\cos\theta\): Represents the x-component of the unit circle corresponding to an angle \(\theta\).
• Sine \(\sin\theta\): Represents the y-component of the unit circle corresponding to an angle \(\theta\).

These functions arise from the unit circle—a circle with a radius of one centered at the origin—and help in determining projections, such as converting cylindrical to rectangular coordinates using the identities:

• \(x = r \cos\theta\)
• \(y = r \sin\theta\)

Understanding these relationships is crucial for smoothly moving between different coordinate systems.

Coordinate Systems

Coordinate systems like cylindrical and rectangular ones provide frameworks for referencing points in space. Whether to describe simple lines or complex, three-dimensional objects:

• Cylindrical systems often suit situations where symmetry around a central axis is pivotal, such as circles or spiral patterns.
• Rectangular systems represent linear dimensions and are prevalent in everyday applications like maps and blueprints.

Coordinate systems simplify complex environments into manageable references, allowing mathematical representation and manipulation. They enable concretizing abstract numbers into spatial interpretations and make complex calculations feasible and visually intuitive.