Finding Min Value of Sum of Cosine Angles in 3D Space

In summary, the minimum value of the sum of cosine of angles between three unit vectors in three-dimensional space is -3/2. One can achieve this minimum by placing three unit vectors at 120 degrees apart in a plane. However, it is possible to do better in three-dimensional space using Lagrange multipliers. The minimum is achieved when the three unit vectors are coplanar and form an equilateral triangle.
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  • #2
Are you interested in absolute value (easy = 0) or most negative (harder)?
 
  • #3
mathman said:
Are you interested in absolute value (easy = 0) or most negative (harder)?
Thank You Mathman! Actually I am interested to find minimum value not absolute . What I am thinking Geometrically is that the possibility seems by imagining unit vectors inclined at 120 degrees to each other in 3D space.Am I correct?It gives thevalue -3/2.
 
  • #4
gianeshwar said:
Thank You Mathman! Actually I am interested to find minimum value not absolute . What I am thinking Geometrically is that the possibility seems by imagining unit vectors inclined at 120 degrees to each other in 3D space.Am I correct?It gives thevalue -3/2.
You can put 3 vectors at 120 deg. apart in a plane. I haven't looked at the problem in any detail, but I would guess you could do better in 3d.
 
  • #5
In the general case, it's easiest to do the problem in vector fashion. The angle cosines are easy: ## \cos \theta_{12} = n_1 \cdot n_2 ##. For unit vector n relative to unit vectors n1, n2, n3, we get
$$ S = \cos \theta_1 + \cos \theta_2 + \cos \theta_3 = n \cdot n_1 + n \cdot n_2 + n \cdot n_3 = n \cdot (n_1 + n_2 + n_3) $$
We now want to maximize ## S = n \cdot n_{total} ## and we wish to do so with the constraint that ## |n| = 1 ##. This is easiest to do with Lagrange multipliers:
$$ S' = S - \frac12 \lambda (n^2 - 1) $$
Taking derivatives with respect to all components of n, we get ## \lambda n = n_1 + n_2 + n_3 = n_{total} ##, meaning that n is proportional to ntotal.
 
  • #6
It seems to me that -3/2 is indeed the minimum. Denote three unit vectors in 3-space by a,b,c with components a(i), b(i), c(i) where i=1,2,3. Then we wish to minimize a.b + b.c + c.a subject to a^2=b^2=c^2=1. Using Lagrange multipliers L,M,N we wish to minimize:

a(i)b(i) + b(i)c(i) + c(i)a(i) - L(a^2-1) - M(b^2-1) - N(c^2-1) where I have used summation convention for repeated indices.

Differentiating by a(i), b(i), c(i) respectively we get

b(i) + c(i) =2La(i)
a(i) + c(i) =2Mb(i)
b(i) + a(i) =2Nc(i)

it is immediately apparent from this that all the vectors are coplanar. furthermore, subtracting the first two of these gives
b(i) (1+2M) = a(i) (1+2L)

But by inspection the minimum is not achieved by having a(i) and b(i) be the same vector, since the cosine is then maximized, so we must have L=M=N= -1/2

Therefore b(i) + c(i) = -a(i), or a + b + c = 0. Hence at the extremum, a,b,c are at the vertices of an equilateral triangle, as conjectured.
 
  • #7
Thank You Friends!
 

1. What is the purpose of finding the minimum value of the sum of cosine angles in 3D space?

The purpose of finding the minimum value of the sum of cosine angles in 3D space is to determine the most efficient arrangement of vectors in 3D space. This is useful in various fields such as physics, engineering, and computer graphics.

2. How is the minimum value of the sum of cosine angles in 3D space calculated?

The minimum value of the sum of cosine angles in 3D space is calculated by finding the dot product of the vectors and taking the inverse cosine of the result. This value will be the minimum value of the sum of cosine angles in 3D space.

3. What is the significance of the minimum value of the sum of cosine angles in 3D space?

The minimum value of the sum of cosine angles in 3D space represents the most optimal arrangement of vectors in terms of minimizing the overall angle between them. This can help in optimizing designs and improving efficiency in various applications.

4. Can the minimum value of the sum of cosine angles in 3D space be negative?

No, the minimum value of the sum of cosine angles in 3D space cannot be negative. The range of cosine values is between -1 and 1, so the minimum value will always be equal to or greater than -1.

5. How does the number of vectors affect the minimum value of the sum of cosine angles in 3D space?

The number of vectors does not affect the minimum value of the sum of cosine angles in 3D space. The minimum value is solely dependent on the dot product of the vectors, not the number of vectors itself.

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