Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 1: Problem 4
Convert point \((-8,8,-7)\) from Cartesian coordinates to cylindrical coordinates.
Short Answer
Expert verified
Cylindrical coordinates are \(\left(8\sqrt{2}, \frac{3\pi}{4}, -7\right)\).
Step by step solution
01
Understanding the Problem
Our task is to convert Cartesian coordinates (-8, 8, -7) into cylindrical coordinates.Cylindrical coordinates are given as \((r, \theta, z)\), where \(r\) is the radial distance, \(\theta\) is the angle in the x-y plane, and \(z\) is the height which corresponds to the z-value in Cartesian coordinates.
02
Calculate Radial Distance (r)
In cylindrical coordinates, the radial distance \(r\) is calculated using the formula\[ r = \sqrt{x^2 + y^2} \].Given \(x = -8\) and \(y = 8\), we find\[ r = \sqrt{(-8)^2 + 8^2} = \sqrt{64 + 64} = \sqrt{128} = 8\sqrt{2} \].
03
Find Angle (\(\theta\))
The angle \(\theta\) can be found using the formula \(\theta = \tan^{-1}(\frac{y}{x})\).Substitute the values \(x = -8\) and \(y = 8\):\[ \theta = \tan^{-1}(\frac{8}{-8}) = \tan^{-1}(-1) \].Since the point is in the second quadrant, \(\theta = \frac{3\pi}{4}\) (as angles are measured from the positive x-axis).
04
Extract Height (z)
In cylindrical coordinates, the height \(z\) remains the same as in Cartesian coordinates.Thus, \(z = -7\).
05
Compile the Cylindrical Coordinates
Now that we have all the components, the cylindrical coordinates are given by \((r, \theta, z)\).Therefore, the cylindrical coordinates of the point (-8, 8, -7) are \(\left(8\sqrt{2}, \frac{3\pi}{4}, -7\right)\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cartesian Coordinates
Cartesian coordinates are a way to specify the location of a point in space using a pair (in 2D) or triplet (in 3D) of numbers.
In a 3D system, these numbers are represented as \(x, y, z\). Here, each of these values describes the point's distance from three perpendicular planes: the x-plane, y-plane, and z-plane.
This system is ideal for representing points in a straightforward, linear manner.
In a 3D system, these numbers are represented as \(x, y, z\). Here, each of these values describes the point's distance from three perpendicular planes: the x-plane, y-plane, and z-plane.
This system is ideal for representing points in a straightforward, linear manner.
- \(x\) represents the horizontal distance from the y-axis.
- \(y\) denotes the vertical distance from the x-axis.
- \(z\) indicates the distance from the x-y plane.
Cylindrical Coordinates
Cylindrical coordinates provide an alternative way to locate points in three-dimensional space, using a combination of length and angle.
They are denoted as \(r, \theta, z\), where \(r\) is the radial distance (a measure of length from the origin in the x-y plane), \(\theta\) is the angle formed with the positive x-axis, and \(z\) remains consistent with its Cartesian counterpart.
They are denoted as \(r, \theta, z\), where \(r\) is the radial distance (a measure of length from the origin in the x-y plane), \(\theta\) is the angle formed with the positive x-axis, and \(z\) remains consistent with its Cartesian counterpart.
- Radial Distance (r): It's akin to the radius of a circle in the plane.
- Angle (\theta): Expressed in radians, indicating direction.
- Height (z): Same as Cartesian, indicating elevation from the x-y plane.
Radial Distance
In the context of cylindrical coordinates, the radial distance \(r\) is the distance from the origin to the point's projection in the x-y plane.
This is calculated using the formula \[ r = \sqrt{x^2 + y^2} \].
The radial distance is essential as it partially determines the point鈥檚 position in the plane.
Consider how:
This is calculated using the formula \[ r = \sqrt{x^2 + y^2} \].
The radial distance is essential as it partially determines the point鈥檚 position in the plane.
Consider how:
- \(r\) acts like the radius of a circle centered at the origin.
- It is always non-negative, capturing the distance without direction.
Angle Calculation
Finding the angle \(\theta\) in cylindrical coordinates requires understanding the trigonometric relation in the x-y plane.
The formula \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \) gives you the angle relative to the positive x-axis.
While the formula appears simple, correctly interpreting \(\theta\) requires attention to the signs and quadrants. Here are steps to consider:
The formula \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \) gives you the angle relative to the positive x-axis.
While the formula appears simple, correctly interpreting \(\theta\) requires attention to the signs and quadrants. Here are steps to consider:
- Identify \(x\) and \(y\)'s signs to determine the quadrant.
- Use inverse tangent carefully, as it typically returns values from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).
- Adjust \(\theta\) based on the quadrant for accurate directional measurement.
