Problem 40 - Advanced Engineering Mathematics

Question

Convert the point given in rectangular cocrdinates to cylindrical coondinates. $$ (2 \sqrt{3}, 2,17) $$

Step-by-Step Solution

Answer

The cylindrical coordinates are \((4, \frac{\pi}{6}, 17)\).

Step 1: Identify Rectangular Coordinates

The given rectangular coordinates are \((x,y,z) = (2\sqrt{3}, 2, 17)\). We will convert these to cylindrical coordinates \((r, \theta, z)\).

Step 2: Find Radius r

Calculate the radius \(r\) using the formula \(r = \sqrt{x^2 + y^2}\). Substitute the given values: \(r = \sqrt{(2\sqrt{3})^2 + 2^2} = \sqrt{12 + 4} = \sqrt{16} = 4\).

Step 3: Determine Angle θ

Calculate the angle \(\theta\) using \(\theta = \tan^{-1}(\frac{y}{x})\). For \((x,y)=(2\sqrt{3},2)\), find \(\theta = \tan^{-1}(\frac{2}{2\sqrt{3}}) = \tan^{-1}(\frac{1}{\sqrt{3}}) = \frac{\pi}{6}\) radians.

Step 4: Use Given z Coordinate

In cylindrical coordinates, the \(z\) value remains the same as in rectangular coordinates. Thus, \(z = 17\).

Step 5: Compile Cylindrical Coordinates

Now that we have \(r = 4\), \(\theta = \frac{\pi}{6}\), and \(z = 17\), the cylindrical coordinates are \((4, \frac{\pi}{6}, 17)\).

Key Concepts

Rectangular to Cylindrical Conversion

Understanding how to convert from rectangular (also known as Cartesian) to cylindrical coordinates is an essential skill when dealing with 3D space in mathematics and physics. Rectangular coordinates are expressed as

• \((x, y, z)\)

which are based on the typical Cartesian plane that many students are familiar with. However, cylindrical coordinates are given as

• \((r, \theta, z)\)

where:

• \(r\) is the radial distance from the z-axis,
• \(\theta\) (theta) is the angle in the xy-plane measured from the positive x-axis,
• \(z\) remains the same as it is the height in both systems.

To convert from rectangular to cylindrical coordinates, you need to determine the radial distance using the formula:

• \( r = \sqrt{x^2 + y^2} \)

Then, calculate the angle \(\theta\) using the inverse tangent function:

• \( \theta = \tan^{-1}(\frac{y}{x}) \)

Finally, the height \(z\) remains unchanged. Applying these conversions allows you to work with cylindrical coordinates comfortably.

Coordinate Systems

Coordinate systems are frameworks that enable us to specify points in space in a clear and mathematical way. In 3D space, some of the most common coordinate systems include

• rectangular coordinates,
• cylindrical coordinates,
• and spherical coordinates.

Each system has its advantages and applications:

• Rectangular coordinates are ideal for problems involving straight lines or when analyzing planar surfaces.
• Cylindrical coordinates are extremely useful in scenarios involving circular or symmetrical features, like tubes or cylindrical objects.
• Spherical coordinates are beneficial when dealing with objects that have a circular symmetry, like spheres.

Cylindrical coordinates provide a blend of rectangular and polar coordinates, using radius \(r\) and angle \(\theta\) in the xy-plane combined with a height \(z\). Understanding which coordinate system to use can simplify problems and is crucial in fields like physics and engineering.

Trigonometry in Coordinate Conversion

Trigonometry plays a key role in converting between different coordinate systems, especially when moving from rectangular to cylindrical coordinates. The angle \(\theta\) in cylindrical coordinates, found using trigonometric functions, determines the point's direction in the xy-plane.

When converting coordinates, use the tangent function to find \(\theta\) because it relates the y and x values of a point. The formula

• \( \theta = \tan^{-1}(\frac{y}{x}) \)

helps ascertain the angle from the positive x-axis to the point in the xy-plane.

Using trigonometry, especially inverse trigonometric functions for angles, ensures precision in conversion steps. This validates the calculated cylindrical coordinates by making sure that any angle-specific calculations are accurate and align correctly with the expected geometrical representation. Learning how to seamlessly use trigonometry for these conversions boosts your ability to tackle a wide array of mathematical and physical challenges.