The points (2, 3,6) and (6, 3, !2) both lie on the sphere x
2
+ y
2
+ z
2
= 49. What is the shortest
path between them if travel is restricted to the surface of the sphere?
The shortest path will be along a great circle along the surface, which will be an arc with radius 7, the radius of the sphere. The length of the arc will be 7 theta, where theta (in radians) is the angle between two lines from the center to the two points on the sphere. Calculate that angle (theta) and you will have it. You can do it with a vector cross product.
7*7 sin theta = |R1 x R2|
R1 = 2i + 3j + 6k
R2 = 6i + 3j - 2k
etc
I will leave the rest to you. Perhaps another teacher has an easier way, but this looks like a good way to do it.
The second point should be -2, just so you know
To find the shortest path between the given points on the surface of a sphere, you'll first need to find the equation of the great circle that passes through these points.
Here's how you can do it step-by-step:
1. Start with the equation of the sphere:
x^2 + y^2 + z^2 = 49
2. Replace 'x' with '2', 'y' with '3', and 'z' with '6' (since (2, 3, 6) lies on the sphere):
2^2 + 3^2 + 6^2 = 49
4 + 9 + 36 = 49
49 = 49
Hence, the point (2, 3, 6) satisfies the equation of the sphere.
3. Similarly, replace 'x' with '6', 'y' with '3', and 'z' with '-2' (since (6, 3, -2) lies on the sphere):
6^2 + 3^2 + (-2)^2 = 49
36 + 9 + 4 = 49
49 = 49
Hence, the point (6, 3, -2) also satisfies the equation of the sphere.
4. Now, we have two points that lie on the surface of the sphere. We can find the equation of the great circle by finding the normal vector to the plane containing these points.
5. Find the vector between the two points by subtracting the coordinates of the first point from the second point:
<6 - 2, 3 - 3, -2 - 6> = <4, 0, -8>
6. Normalize the vector by dividing it by its magnitude:
||<4, 0, -8>|| = sqrt(4^2 + 0^2 + (-8)^2) = sqrt(80) = 4√5
So, the normalized vector is: <4/4√5, 0/4√5, -8/4√5> = <1/√5, 0, -2/√5> = <√5/5, 0, -2√5/5>
7. The normalized vector is the direction vector of the normal to the plane. So, the equation of the plane passing through the two points is:
√5/5 (x - 2) + 0 (y - 3) - 2√5/5 (z - 6) = 0
Simplifying, you get: √5 (x - 2) - 2√5 (z - 6) = 0
8. This plane is the same as the great circle passing through the two points on the sphere. And the shortest path between the two points on the surface of the sphere is the arc length along this great circle.
So, the equation of the great circle on the surface of the sphere is √5 (x - 2) - 2√5 (z - 6) = 0.