Reflection Equation for Sphere in (x-1)^2+(y+2)^2+(z-4)^2=16 at (2,1,-2)

In summary, the equation of reflection of the sphere in (x-1)^2+(y+2)^2+(z-4)^2=16 with respect to the point (2,1,-2) is (x-3)^2+(y-4)^2+(z+8)^2=16. This is found by finding the midpoint between the center of the sphere and the reflection point, which is (3,4,-8) in this case. This process is related to calculus because it involves solving equations and finding the distance between points, which are concepts used in calculus.
  • #1
bobsmiters
12
0
Find the equation of reflection of the sphere in (x-1)^2+(y+2)^2+(z-4)^2=16 with respect to the point (2,1,-2).

There was another question asking for the reflection equation but it was with respect to the xy-plane so it just meant changing some signs. What I came up with for this question was

(x-3)^2+(y-4)^2+(z+8)^2=16 ?
 
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  • #2
Yes, that is correct.

"Reflection in a point", q, means that the image of point p is on the line from p through q and the same distance from q as p. That is, the "reflection point" must be the the midpoint of the segment between a point and its image. The center of the sphere is at (1, -2, 4). If we take (a, b, c) as the image, then we must have (1+ a)/2= 2, (-2+ b)/2= 1, and (c+ 4)/2= -2. Solving those equations gives a=3, b= 4, c= -8 so your equation is correct. (Of course, the image circle has the same radius as the original.)
 
  • #3
what exactly does this have to do with calculus?
 

FAQ: Reflection Equation for Sphere in (x-1)^2+(y+2)^2+(z-4)^2=16 at (2,1,-2)

What is Multivariable Calculus?

Multivariable Calculus is a branch of mathematics that deals with functions of multiple variables. It involves the study of functions that have more than one independent variable, and the techniques used to analyze and manipulate them.

Why is Multivariable Calculus important?

Multivariable Calculus is important because it is used to solve real-world problems in fields such as physics, engineering, economics, and statistics. It also provides a foundation for further study in advanced mathematics and other scientific disciplines.

What are some applications of Multivariable Calculus?

Multivariable Calculus has many applications, including optimization, curve fitting, differential equations, and vector calculus. It is also used in fields such as computer graphics, robotics, and machine learning.

What are the key concepts in Multivariable Calculus?

The key concepts in Multivariable Calculus include partial derivatives, multiple integrals, vector-valued functions, and the gradient, divergence, and curl operators. Other important topics include line and surface integrals, Taylor series, and the Fundamental Theorem of Calculus.

What are some common challenges in learning Multivariable Calculus?

Some common challenges in learning Multivariable Calculus include visualizing and understanding functions in more than two dimensions, mastering the various integration techniques, and applying the concepts to real-world problems. It is also important to have a strong foundation in single-variable calculus before tackling Multivariable Calculus.

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