Problem 35 - Advanced Engineering Mathematics

Question

In Problems \(35-38\), convert the point given in cylindrical coardinates 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 the given cylindrical coordinates

The cylindrical coordinates are given as \((10, \frac{3\pi}{4}, 5)\). This means: \(r = 10\), \(\theta = \frac{3\pi}{4}\), and \(z = 5\).

Step 2: Convert the cylindrical coordinate to rectangular coordinate for the x-component

The formula to convert the radial coordinate \(r\) and the angle \(\theta\) to the rectangular x-component is \(x = r \cos\theta\). Substitute the known values: \[ x = 10 \cos\left(\frac{3\pi}{4}\right) = 10 \times \left(-\frac{\sqrt{2}}{2}\right) = -5\sqrt{2}. \]

Step 3: Convert the cylindrical coordinate to rectangular coordinate for the y-component

The formula to convert \(r\) and \(\theta\) to the rectangular y-component is \(y = r \sin\theta\). Substitute the known values: \[ y = 10 \sin\left(\frac{3\pi}{4}\right) = 10 \times \frac{\sqrt{2}}{2} = 5\sqrt{2}. \]

Step 4: Translate the z-component directly to rectangular coordinates

The \(z\)-component in cylindrical coordinates remains the same in rectangular coordinates. Therefore, \(z = 5\).

Step 5: Compile the rectangular coordinates

Now gather all computed components. The rectangular coordinates are \((-5\sqrt{2}, 5\sqrt{2}, 5)\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are defined by three axes: x, y, and z. In a 3-dimensional space, these axes are perpendicular to each other. Each point in this space can be represented by a set of three values which specify a position relative to the origin.

• The x-component represents the horizontal distance from the y-z plane.
• The y-component indicates the vertical distance from the x-z plane.
• The z-component shows the height from the x-y plane.

Rectangular coordinates are particularly useful in describing locations in everyday environments, like mapping when driving directions or locating objects in a room.

Cylindrical Coordinates

Cylindrical coordinates are a three-dimensional extension of polar coordinates with an added height component. Instead of being defined by x, y, and z axes directly, a point is determined by:

• The radial distance (\( r \)) from the origin in the x-y plane.
• The angular coordinate (\( \theta \)), representing the counterclockwise angle from the positive x-axis.
• The vertical position (\( z \)) from the x-y plane, similar to the z-coordinate in rectangular values.

To convert from cylindrical to rectangular coordinates, use the following formulas:

• \( x = r \cos(\theta) \)
• \( y = r \sin(\theta) \)
• \( z = z \) remains the same

Understanding this system is helpful for tasks involving rotational symmetry, like analyzing positions around a cylindrical object.

Trigonometric Functions

Trigonometric functions are mathematical relationships involving angles, most commonly used to relate angles to side lengths in right triangles. They are also essential when dealing with coordinate transformations involving angles, such as converting cylindrical coordinates to rectangular coordinates.

The primary trigonometric functions include:

• Cosine (\( \cos \)): Calculates the adjacent side's length divided by the hypotenuse for an angle in a right triangle. Used to find the x-component in conversions.
• Sine (\( \sin \)): Represents the opposite side's length over the hypotenuse. It helps determine the y-component during coordinate conversion.

For the given problem, knowing the angle \( \frac{3\pi}{4} \)is crucial. You need cosine and sine to compute the x and y components respectively, thus transforming cylindrical coordinates into a more familiar rectangular form.