Problem 38 - Advanced Engineering Mathematics

Question

In Problems \(35-38\), convert the point given in cylindrical coardinates 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: Identify Cylindrical Coordinates

Cylindrical coordinates are given in the form \[(r, \theta, z)\]. Here, the point given is \[(4, \frac{7\pi}{4}, 0)\].\(r = 4\) is the radial distance, \(\theta = \frac{7\pi}{4}\) is the angle measured from the positive x-axis, and \(z = 0\) is the height.

Step 2: Convert to Rectangular Coordinates

To convert a point from cylindrical \((r, \theta, z)\) to rectangular coordinates \((x, y, z)\), we use the following equations:- \(x = r \cdot \cos \theta\) - \(y = r \cdot \sin \theta\) - \(z = z\)Substituting the given values:\[x = 4 \cdot \cos\left(\frac{7\pi}{4}\right)\]\[y = 4 \cdot \sin\left(\frac{7\pi}{4}\right)\]\[z = 0\]

Step 3: Calculate x-coordinate

The cosine of \(\frac{7\pi}{4}\) is \(\frac{1}{\sqrt{2}}\) or equivalently, \(\cos 45^\circ\) is positive in the fourth quadrant. \[x = 4 \cdot \frac{1}{\sqrt{2}} = 4 \cdot \frac{\sqrt{2}}{2} = 2\sqrt{2}\]

Step 4: Calculate y-coordinate

The sine of \(\frac{7\pi}{4}\) is \(-\frac{1}{\sqrt{2}}\) or equivalently, \(\sin 45^\circ\) is negative in the fourth quadrant. \[y = 4 \cdot \left(-\frac{1}{\sqrt{2}}\right) = 4 \cdot \left(-\frac{\sqrt{2}}{2}\right) = -2\sqrt{2}\]

Step 5: Compile Full Solution

Now that we have calculated all components, the rectangular coordinates are:\[(x, y, z) = (2\sqrt{2}, -2\sqrt{2}, 0)\]

Key Concepts

Coordinate Conversion

The process of converting points from one coordinate system to another is known as coordinate conversion. This is an essential mathematical tool used in various fields like physics, engineering, and computer graphics. In this exercise, we're converting cylindrical coordinates to rectangular coordinates.

The main goal here is to transform a point given in terms of \(r, \theta, z\) (cylindrical) to \(x, y, z\) (rectangular). This allows us to utilize the point's information in more familiar Cartesian spaces.

• Cylindrical coordinates are centered around a polar-like system combined with height information (z-axis).
• Rectangular coordinates are the Cartesian coordinates, defined by x, y, and z axes forming a right-angled 3D coordinate system.

Through specific equations such as \(x = r \cdot \cos \theta\) and \(y = r \cdot \sin \theta\), we can accurately locate a point on a Euclidean plane based on its radial distance, angular placement, and height. This paramount understanding helps interpret spatial data effectively in diverse applications.

Cylindrical Coordinates

Cylindrical coordinates are used to define a point in a 3D space using three parameters: radial distance ( ), angular coordinate ( heta), and height (z). This system is especially useful for objects or situations that have rotational symmetry around an axis.

Typically, cylindrical coordinates appear as \(r, \theta, z\):

• is the radial distance from the z-axis, similar to the radius in a polar coordinate system.
• \(\theta\) is the angular position, rotating around the z-axis, measured from the positive x-axis.
• \(z\) is the vertical position, which is a straight linear measure along the z-axis.

In this exercise, the cylindrical coordinates given were \(4, \frac{7\pi}{4}, 0\), where \(r = 4\), \(\theta = \frac{7\pi}{4}\), and \(z = 0\).
The representation simplifies the mathematics in scenarios involving rotational motion or sections of wider circular pathways.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are widely used to specify the location of a point in a 3D space through three perpendicular axes: x, y, and z. These coordinates offer a straightforward and intuitive way of understanding spatial dimensions.

In the rectangular system:

• \(x\) represents the horizontal position along the x-axis.
• \(y\) depicts the location along the y-axis, perpendicular to x.
• \(z\) shows the height, projecting straight up from the xy-plane.

From the cylindrical coordinates \(4, \frac{7\pi}{4}, 0\), we derived the rectangular coordinates through:

- \(x = r \cdot \cos \theta = 2\sqrt{2}\)
- \(y = r \cdot \sin \theta = -2\sqrt{2}\)
- \(z = 0\)

Thus, the point in rectangular coordinates is \( (2\sqrt{2}, -2\sqrt{2}, 0)\). These conversions allow for the comparison and integration of geometric data across different coordinate systems, crucial in multivariable mathematics and computational simulations.