Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 9: Problem 66
Convert the points given in rectangular coordinates to spherical coordinates. $$ \left(-\frac{\sqrt{3}}{2}, 0,-\frac{1}{2}\right) $$
Short Answer
Expert verified
Spherical coordinates are \((1, \pi, \frac{2\pi}{3})\).
Step by step solution
01
Identify the Rectangular Coordinates
The given rectangular coordinates in the problem are: \( x = -\frac{\sqrt{3}}{2}, y = 0, z = -\frac{1}{2} \). These are the values we'll convert to spherical coordinates.
02
Calculate the Radial Distance \( r \)
The radial distance \( r \) is calculated using the formula for the magnitude of the vector: \[ r = \sqrt{x^2 + y^2 + z^2} \]Substitute the values:\[ r = \sqrt{\left(-\frac{\sqrt{3}}{2}\right)^2 + 0^2 + \left(-\frac{1}{2}\right)^2} = \sqrt{\frac{3}{4} + \frac{1}{4}} = \sqrt{1} = 1 \]
03
Calculate the Azimuthal Angle \( \theta \)
The azimuthal angle \( \theta \) is measured from the positive x-axis in the xy-plane. Since \( y = 0 \), it simplifies the formula:\[ \theta = \arctan\left(\frac{y}{x}\right) \]However, since \( x = -\frac{\sqrt{3}}{2} \) and \( y = 0 \), \( \theta \) is either \( 0 \) or \( \pi \). Since the point is on the negative x-axis, \( \theta = \pi \).
04
Calculate the Polar Angle \( \phi \)
The polar angle \( \phi \) is calculated from the positive z-axis:\[ \phi = \arccos\left(\frac{z}{r}\right) \]Substitute the given values: \[ \phi = \arccos\left(\frac{-\frac{1}{2}}{1}\right) = \arccos(-\frac{1}{2}) = \frac{2\pi}{3} \]
05
Write the Spherical Coordinates
The spherical coordinates \( (r, \theta, \phi) \) are now summarized from our calculations:\[ (r, \theta, \phi) = (1, \pi, \frac{2\pi}{3}) \]
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.
Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, are a way to specify a point in a three-dimensional space using three values: x, y, and z. Think of it as a grid system, much like the one used on a map, that helps locate a place.
- The x-coordinate tells you how far to move horizontally from the origin, along the x-axis.
- The y-coordinate shows the vertical movement along the y-axis.
- The z-coordinate indicates movement up or down along the z-axis, adding depth to the position.
Radial Distance Calculation
The radial distance, denoted as \( r \), is a measure of how far a point is from the origin, regardless of direction. It's essentially the length of the vector from the origin to the point, much like a ruler's reading tells you the length of a line.
- This distance is calculated using the formula \( r = \sqrt{x^2 + y^2 + z^2} \).
- It combines the effects of all three coordinates into a single distance.
Azimuthal Angle Theta
The azimuthal angle \( \theta \) is measured in the xy-plane from the positive x-axis. It is essentially like determining the direction of the point on the horizontal plane, similar to how a compass shows direction on a map.
- \( \theta \) is found using \( \theta = \arctan\left(\frac{y}{x}\right) \).
- When \( y = 0 \), this simplifies the situation considerably.
Polar Angle Phi
The polar angle \( \phi \) is taken from the positive z-axis down to the line formed by the point and the origin. Picture it like looking at how high or low the point sits in space relative to the ground.
- \( \phi \) is calculated using \( \phi = \arccos\left(\frac{z}{r}\right) \).
- It helps translate the height component of the point relative to the origin.
