Orthogonal Vectors for Sphere Construction: Find Center and Radius

In summary, orthogonal vectors are crucial in sphere construction as they aid in finding the center and radius of the sphere. They can be used to construct spheres of any size by locating three points on the sphere's surface and drawing two orthogonal vectors from the center to two of those points. The radius of the sphere can be calculated using the formula r = √(a² + b² + c²), where a, b, and c are the lengths of the orthogonal vectors. It is not possible to use more than three orthogonal vectors to construct a sphere as three non-collinear points are enough to uniquely define a sphere in three-dimensional space.
  • #1
nameVoid
241
0
R <x,y,z>
A<a1,a2,a3>
B<b1,b2,b3>

Show that (r-a).(r-b)=0 represents a sphere find its center and radius

So i see that if 2 vectors are orthogonal you can create a sphere and find the radius and center but can somone better explain this problem
 
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  • #2
What is there to explain? Write out the equations and put them in standard form to identify it. Geometrically, you may recall that an angle inscribed in a semicircle is a right angle. This is the 3D analogue of that.
 

Related to Orthogonal Vectors for Sphere Construction: Find Center and Radius

1. What is the significance of orthogonal vectors in sphere construction?

Orthogonal vectors are essential in sphere construction because they help determine the center and radius of the sphere. These vectors are perpendicular to each other and provide important information about the shape and dimensions of the sphere.

2. How do you find the center of a sphere using orthogonal vectors?

To find the center of a sphere using orthogonal vectors, you need to first locate three points on the sphere's surface. Then, draw two orthogonal vectors from the center of the sphere to two of the points. The intersection of these two vectors will be the center of the sphere.

3. Can orthogonal vectors be used to construct spheres of any size?

Yes, orthogonal vectors can be used to construct spheres of any size. As long as you have three points on the sphere's surface and draw two orthogonal vectors from the center to two of those points, you can determine the center and radius of the sphere regardless of its size.

4. Is there a specific formula for calculating the radius of a sphere using orthogonal vectors?

Yes, the formula for calculating the radius of a sphere using orthogonal vectors is r = √(a² + b² + c²), where a, b, and c are the lengths of the orthogonal vectors drawn from the center to the three points on the sphere's surface.

5. Can you use more than three orthogonal vectors to construct a sphere?

No, it is not possible to use more than three orthogonal vectors to construct a sphere. In three-dimensional space, three non-collinear points are enough to uniquely define a sphere, and adding more points or vectors would create redundant information.

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