Problem 39 - Advanced Engineering Mathematics

Question

In Problems \(39-42\), convert the point given in rectangular coardinates to cylindrical coordinates. $$ (1,-1,-9) $$

Step-by-Step Solution

Answer

\((\sqrt{2}, -\frac{\pi}{4}, -9)\)

Step 1: Understand the Conversion Formulas

For converting rectangular coordinates \((x, y, z)\) to cylindrical coordinates \((r, \theta, z)\), use the formulas:\[ r = \sqrt{x^2 + y^2} \] for the radial distance, and \[ \theta = \arctan \left( \frac{y}{x} \right) \] for the angle in radians. The \(z\)-coordinate remains the same.

Step 2: Compute the Radial Distance \(r\)

Substitute \(x = 1\) and \(y = -1\) into the formula for \(r\): \[ r = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}. \] Thus, \(r = \sqrt{2}\).

Step 3: Calculate the Angle \(\theta\)

Use the \(\arctan\) formula to find \(\theta\): \[ \theta = \arctan \left( \frac{-1}{1} \right) = \arctan(-1). \] The angle \(\arctan(-1)\) is in the fourth quadrant, and equals \(-\frac{\pi}{4}\) radians.

Step 4: Determine the \(z\)-Coordinate

The \(z\)-coordinate remains the same in cylindrical coordinates. Therefore, \(z = -9\).

Step 5: Write Cylindrical Coordinates

Combine the calculated \(r\), \(\theta\), and \(z\) to express the point in cylindrical coordinates: \(\left( \sqrt{2}, -\frac{\pi}{4}, -9 \right)\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way of expressing points in a plane or space using two or three dimensions. This system utilizes axes at right angles to each other, typically labeled as the x, y, and z axes. Each point in this coordinate system is defined by an ordered pair or triple of values that indicate positions relative to these axes.

• The x-coordinate specifies the point's horizontal position.
• The y-coordinate gives the vertical position.
• The z-coordinate is used in three-dimensional space to denote depth.

In a 3D rectangular coordinate system, each point is represented as \(x, y, z\). For example, in the problem \( (1, -1, -9) \), the point is located one unit along the positive x-axis, one unit along the negative y-axis, and nine units along the negative z-axis.
This coordinate system forms the foundation for many mathematical computations, making it an essential tool for students studying physics, engineering, and computer graphics.

Coordinate Conversion

Coordinate conversion is the process of changing coordinates from one system to another. This is particularly useful in various fields like navigation, architecture, and mathematics, as different problems are easier to solve in different coordinate systems. Converting from rectangular coordinates to cylindrical coordinates is a common task.
To convert rectangular coordinates \(x, y, z\) to cylindrical coordinates \(r, \theta, z\), the following steps are used:

• Radial Distance (r): This is calculated using the formula \(r = \sqrt{x^2 + y^2}\), which determines the distance from the z-axis to the point in question.
• Angle (\( \theta \)): The angle is found using \(\theta = \arctan\left(\frac{y}{x}\right)\). This is the angle the point makes with the positive x-axis in the xy-plane.
• Height (z): This value remains unchanged during the conversion, as cylindrical and rectangular coordinates share the same height component.

These formulas allow us to easily map points from the rectangular system to the cylindrical system, offering a different perspective that can simplify calculation and understanding of spatial relationships.

Trigonometry

Trigonometry is a branch of mathematics that studies relationships between the angles and lengths of triangles. It plays a significant role in coordinate conversion, particularly when determining the angle \(\theta\) in various coordinate systems.
In our context of converting rectangular coordinates to cylindrical coordinates, trigonometry helps determine the direction of the radial distance within the plane. The concept of arctangent (arctan) is crucial here.

• Understanding Arctan: The function \(\arctan\left(\frac{y}{x}\right)\) calculates the angle whose tangent is \(\frac{y}{x}\). It returns values in radians, which is a standard unit of angular measure in mathematics.
• Handling Radian Measures: Knowing that angles are measured from the positive x-axis in radians is key, especially since calculations for \(-1 < \theta < 0\) fall in the negative quadrant.

Using trigonometry and understanding of these functions is crucial for accurately converting coordinates and ensuring spatial relationships are preserved across different coordinate systems.