Proving cos^2(\theta_x)+cos^2(\theta_y)+cos^2(\theta_z)=1

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In summary: Oh well.In summary, the problem asks you to find the sum of the cosines squared of the angles between a vector and the three axes. Using either geometry or vector algebra, you can find this sum if you treat the vector's components in relation to each axis.
  • #1
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Homework Statement


Prove that sum of the cosines squared of the angles between a vector and the x, y, and z axes equals 1. Prove using either geometry or vector algebra.

Homework Equations


[itex]cos^2(\theta_x)+cos^2(\theta_y)+cos^2(\theta_z)=1[/itex]

The Attempt at a Solution


I started by trying to pull out the angles with
[itex]\cos^{-1}[/itex]
Getting
[itex]\theta_x+\theta_y+\theta_z=0+2k\pi[/itex]
(Edit: This is no longer valid)
This doesn't make sense to me since I know that the maximum sum of the angles can only be around 180 degrees (I think). So I thought maybe the geometric approach would have something to do with triangles. But I don't know how to show that analytically.
(Edit: Still clueless)

I'm also lost on how to start with the vector algebra. I'm looking over the identities of vectors now but to no avail.

Thoughts?
 
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  • #2
Are you sure the problem doesn't say the sum of the squares of the cosines?
 
  • #3
Aaaah, edited :D

Edit: I sort of haven't looked at the book in a while and have been banging my head so hard I must have forgotten that important part :D
 
  • #4
Hmm, so let's see, if there's 3 triangles between the vector and the axes, you can use Pythagorean theorem to get the cosine squared of each component right?
 
  • #5
Cake said:
Hmm, so let's see, if there's 3 triangles between the vector and the axes, you can use Pythagorean theorem to get the cosine squared of each component right?
I don't quite follow. Provide more details, please (in the form of equations).

Chet
 
  • #6
What I'm saying is if you treat the components of the vector in relation to each axis, you can break the components into [itex]A cos(\theta_x), A cos(\theta_y), A cos(\theta_z)[/itex] right? That's how you would relate the vectors position in relation to the axes to cosine?
 
  • #7
Cake said:
What I'm saying is if you treat the components of the vector in relation to each axis, you can break the components into [itex]A cos(\theta_x), A cos(\theta_y), A cos(\theta_z)[/itex] right? That's how you would relate the vectors position in relation to the axes to cosine?
Good so far. what next?

Chet
 
  • #8
Can I imagine that this is 2 dimensional instead of 3 and put the vector on a 2D plane to use pythagorean theorem to get them to be squared?
[itex]c^2=A^2 cos^2(\theta_x)+A^2 cos^2(\theta_y)[/itex]

That's how I was visualizing earlier too it, but it gets rid of the z component, so I'm not sure how I can bring the z axis back into the equation.
 
  • #9
Cake said:
Can I imagine that this is 2 dimensional instead of 3 and put the vector on a 2D plane to use pythagorean theorem to get them to be squared?
[itex]c^2=A^2 cos^2(\theta_x)+A^2 cos^2(\theta_y)[/itex]

That's how I was visualizing earlier too it, but it gets rid of the z component, so I'm not sure how I can bring the z axis back into the equation.
The left side of this equation should be A2. Does this give you a hint as to what to do in 3D?

Chet
 
  • #10
The Pythagorean identity for 3D vectors is [itex]A^2 = A_x^2+A_y^2+A_z^2[/itex] but I didn't think it would be that easy that I could just use that willy nilly for this problem. But it makes sense. So:

[itex]A^2 = A^2 cos^2(\theta_x)+A^2 cos^2(\theta_y)+A^2cos^2(\theta_z)[/itex]

Divide by [itex]A^2[/itex]

Aaaaand finished.

Still not sure why I didn't get that originally.
 

1. What does the equation cos^2(\theta_x)+cos^2(\theta_y)+cos^2(\theta_z)=1 mean?

The equation represents the Pythagorean theorem in three dimensions. It states that the sum of the squares of the cosine of the angles in a right triangle is equal to 1.

2. How can this equation be proven?

This equation can be proven using trigonometric identities and the Pythagorean theorem for both the x and y coordinates, and then combining them together to get the desired result.

3. What is the significance of this equation in mathematics?

This equation is significant in mathematics because it shows the relationship between the lengths of the sides and the angles in a right triangle in three dimensions. It is also a fundamental concept in trigonometry and has various applications in fields such as physics, engineering, and geometry.

4. Can this equation be applied to non-right triangles?

No, this equation only applies to right triangles. In non-right triangles, the sum of the squares of the cosine of the angles will not equal 1.

5. How is this equation used in real-world scenarios?

This equation is used in real-world scenarios to calculate unknown angles or sides in a right triangle, such as in navigation, surveying, and construction. It is also used in physics to analyze forces and motion in three dimensions.

Suggested for: Proving cos^2(\theta_x)+cos^2(\theta_y)+cos^2(\theta_z)=1

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