Problem 35 - Advanced Engineering Mathematics

Question

Convert the point given in cylindrical cocudinates to rectangular cocrdinates. $$ \left(10, \frac{3 \pi}{4}, 5\right) $$

Step-by-Step Solution

Answer

The rectangular coordinates are \((-5\sqrt{2}, 5\sqrt{2}, 5)\).

Step 1: Identify Cylindrical Coordinates

The point given in cylindrical coordinates is \((r, \theta, z) = (10, \frac{3\pi}{4}, 5)\). This means that the radial distance \(r\) is 10, the angular coordinate \(\theta\) is \(\frac{3\pi}{4}\) radians, and the height \(z\) is 5.

Step 2: Understand Conversion Formulas

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

Step 3: Calculate x-coordinate

Substitute the given values into the formula for \(x\): \[ x = 10 \cos\left(\frac{3\pi}{4}\right) = 10 \times \left(-\frac{\sqrt{2}}{2}\right) = -5\sqrt{2}. \]

Step 4: Calculate y-coordinate

Substitute the given values into the formula for \(y\): \[ y = 10 \sin\left(\frac{3\pi}{4}\right) = 10 \times \frac{\sqrt{2}}{2} = 5\sqrt{2}. \]

Step 5: Identify z-coordinate

The \(z\)-coordinate remains the same in rectangular coordinates. Thus, \(z = 5\).

Step 6: Write Rectangular Coordinates

The rectangular coordinates are \((-5\sqrt{2}, 5\sqrt{2}, 5)\).

Key Concepts

Coordinate Systems

In the world of mathematics and physics, coordinate systems are vital for pinpointing the location of points in both two-dimensional and three-dimensional spaces. These systems allow us to assign a unique set of numbers, known as coordinates, to each point to accurately measure and communicate their position.

The two main types of coordinate systems often used are Cartesian (rectangular) and cylindrical systems. Each of these systems has unique properties and is used in different scenarios. Understanding their differences and applications is essential for various fields like engineering, physics, and computer graphics.

While Cartesian coordinates describe points with a pair or triplet of numerical values representing distances along axes (such as x, y, and z), cylindrical coordinates involve a hybrid of radial distances, angles, and heights.

Conversion Formulas

Converting between different coordinate systems involves using specific formulas. These formulas help in translating from one set of coordinates to another seamlessly. In our context, we focus on the conversion between cylindrical and rectangular coordinates.

The key formulas for converting cylindrical coordinates \( (r, \theta, z) \) to rectangular coordinates \( (x, y, z) \) are as follows:

• For the x-coordinate: \( x = r \cos \theta \)
• For the y-coordinate: \( y = r \sin \theta \)
• The z-coordinate: remains unchanged, so \( z = z \)

These formulas arise from the geometric relationships between the two coordinate systems, linking the aspects of distance, angle, and height. Mastering these equations is crucial for performing accurate conversions.

Cylindrical Coordinates

Cylindrical coordinates are particularly useful in scenarios where problems have some degree of rotational symmetry about an axis. This coordinate system can be visualized as an extension of the polar coordinate system into three dimensions.

In this system, each point is represented by three values:

• Radial distance (r): This signifies how far the point is from the origin as projected onto a base plane.
• Angular coordinate (\(\theta\)): Measured in radians or degrees, it represents the counterclockwise angle from a reference line (typically the positive x-axis).
• Height (z): This is simply the vertical height above or below the base plane.

Understanding how to interpret these coordinates provides the foundation for making conversions to other systems like rectangular coordinates.

Rectangular Coordinates

Rectangular coordinates, often known as Cartesian coordinates, provide a straightforward way of specifying the position of points using a grid of perpendicular lines. In three dimensions, they are expressed as \( (x, y, z) \). Here, each coordinate measures the point’s distance from each of the three perpendicular planes.

• x-coordinate: The distance along the horizontal axis.
• y-coordinate: The distance along the vertical axis.
• z-coordinate: Represents the height in the three-dimensional space.

This system is preferred for its simplicity and ease of visualization. Whether it's plotting a graph in math class or positioning elements in a computer simulation, knowing how to work with rectangular coordinates is highly beneficial. The real power, however, lies in understanding how these coordinates connect and convert to other systems like cylindrical coordinates to solve complex problems.