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Alona [7]
3 years ago
14

Consider a sphere of radius R > 0. Take two antipodal points A and B on the sphere, and another point C on the sphere. Check

that the angle ACB is a right angle.
Use only vectors. No solution involving classical geometry or coordinates will be accepted.

You may want to draw a picture, and introduce a good coordinate system.

Mathematics
1 answer:
aleksklad [387]3 years ago
5 0

Answer:

To this question; we want to show that if two antipodal points on a sphere were A and B, for any random point C on the sphere, AC is perpendicular to BC?

Step-by-step explanation:

I would proceed by imagining the sphere in a three dimensional Cartesian coordinate system. For example, use a sphere of diameter 2 and let it sit at the origin. Then (0, 0, 1) and (0, 0, -1) are the vector locations of the “north and south poles” of the sphere.

Now choose the vector location of any point on the surface of the sphere. It will have a vector location - we’ll call it (x, y, z). Now the vectors from your point to the two poles are (-x, -y, 1-z) and (-x, -y, -1-z).

Now just form the dot product of those two vectors:

(-x, -y, 1-z) . (-x, -y, -1-z) = x^2 +y^2 + (1-z)*(-1-z)

Now the truth of your claim will be embodied in that dot product being zero:

x^2 + y^2 - (1+z)(1-z) = 0

x^2 + y^2 - (1-z^2) = 0

x^2 + y^2 + z^2 = 1

But that last line is just the definition of points on the surface of a sphere of radius 1, so the claim is proven.

Since R>0 and AC is perpendicular to BC, the <ACB is at right angle.

Please, find attached a simple image to show antipodal points (Two points that makes a diameter) and a point C on the sphere.

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Answer:

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Step-by-step explanation:

Let be "x"  the number of hours worked on a project and "y" the total amount Luigi charges for a project.

Based on the data given in the exercise, you can identify that:

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Then, the total amount he charges for a project is the sum of $140 and 25x. This can be represented with the following function:

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To calculate the amount of money earned for 10 hours of work, you must substitute x=10 into the function and evaluate. Then:

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The miles he drove North on Avenue B is 24 miles.

<h3>What is Pythagoras theorem? </h3>

According to the Pythagoras theorem, the square of the hypotenuse is the sum of the square of the opposite sides.

The Pythagoras theorem: a² + b² = c²

Where:

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Please find attached the required diagram. To learn more about Pythagoras theorem, please check: brainly.com/question/14580675

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Step-by-step explanation:

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