Problem 39 - Advanced Engineering Mathematics

Question

Convert the point given in rectangular cocrdinates to cylindrical coondinates. $$ (1,-1,-9) $$

Step-by-Step Solution

Answer

The cylindrical coordinates are \((\sqrt{2}, \frac{7\pi}{4}, -9)\).

Step 1: Understand the Relationship between Coordinates

In rectangular coordinates, we represent points as \((x, y, z)\). In cylindrical coordinates, the point is represented as \((r, \theta, z)\), where \(r\) is the radial distance from the \(z\)-axis, \(\theta\) is the angle in the \(xy\)-plane from the positive \(x\)-axis, and \(z\) remains the same.

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

The radial distance \(r\) is calculated using the formula: \(r = \sqrt{x^2 + y^2}\). For the point \((1, -1, -9)\), substitute \(x = 1\) and \(y = -1\) to get:\[r = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}\.\]

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

The angle \(\theta\) is found using the formula \(\theta = \tan^{-1}(\frac{y}{x})\). For \((1, -1, -9)\), this becomes:\[\theta = \tan^{-1} \left(\frac{-1}{1}\right) = \tan^{-1}(-1)\,\]which gives \(\theta = -\frac{\pi}{4}\) radians (or equivalently \(\theta = \frac{7\pi}{4}\) radians in the range \([0, 2\pi)\), which is often preferred for cylindrical coordinates).

Step 4: Retain the \(z\)-coordinate

In cylindrical coordinates, the \(z\)-coordinate remains unchanged. Therefore, for the given point, \(z = -9\).

Step 5: Assemble the Cylindrical Coordinates

Finally, assemble the cylindrical coordinates using the calculated values: \(r = \sqrt{2}\), \(\theta = \frac{7\pi}{4}\), and \(z = -9\). So the cylindrical coordinates for the point \((1, -1, -9)\) are:\[(\sqrt{2}, \frac{7\pi}{4}, -9)\.\]

Key Concepts

Rectangular Coordinates Conversion

When converting a point from rectangular coordinates to another system like cylindrical coordinates, understanding each component is crucial. A point in rectangular coordinates is presented as

• \((x, y, z)\)

. These

• \(x\)
• \(y\)
• \(z\)

values define the point’s position in a three-dimensional space based on perpendicular axes. It's like locating a specific point in a three-dimensional box.

The process of converting these values into cylindrical coordinates involves three main steps:

• Calculating the radial distance, denoted by \(r\), representing the point’s distance from the \(z\)-axis.
• Finding the angle \(\theta\) in the \(xy\)-plane from the positive \(x\)-axis to where the point lies.
• Leaving the \(z\)-coordinate unchanged, showing the point's height relative to the plane.

Understanding these conversions is essential for translating between systems often used in physics and engineering.

Coordinate Transformation

A coordinate transformation is the process of converting points from one coordinate system to another. This conversion is crucial when analyzing problems in different contexts or simplifying calculations.

In our example, transforming from rectangular to cylindrical coordinates is straightforward. We use specific relationships and formulas to compute the new coordinates:

• The formula \(r = \sqrt{x^2 + y^2}\) calculates the radial distance \(r\) by evaluating the distance of a point from the \(z\)-axis in the \(xy\)-plane.
• The angle \(\theta\) is given by \(\theta = \tan^{-1}(\frac{y}{x})\). This evaluates the angle from the positive \(x\)-axis towards the point in the \(xy\)-plane.
• The \(z\) value remains untouched, showing the vertical position.

By executing these calculations, you seamlessly translate the point's representation, utilizing mathematical tools to move fluidly between different perspectives.

Cylindrical Coordinate System

The cylindrical coordinate system provides a blend of circular and linear measurements, perfectly capturing points that align around a central axis, such as around a pole or tube.

In cylindrical coordinates, a point is expressed as

• \((r, \theta, z)\)

. Here,

• \(r\) signifies the distance from the point to the \(z\)-axis, mirroring the radius of a circle's segment.
• \(\theta\) is the angular rotation from the positive \(x\)-axis, akin to the angle in polar coordinates.
• \(z\) maintains its role in determining the position's height above or below the reference plane.

This system simplifies computations where symmetry around an axis is prevalent, making it indispensable in fields like engineering and physics. For example, while modeling the motion of particles in a circular path or analyzing wave patterns, cylindrical coordinates offer simplicity and insight into the structure and behavior of systems.