Calculating the Mean Value of Vector n_i*n_j in 3D Space - Help and Explanation

AI Thread Summary
The discussion centers on calculating the mean value of the product of components of a random unit vector n in 3D space, specifically < n_i*n_j > for i, j = 1, 2, 3. Participants agree that this mean value results in a 3x3 tensor, as it represents the correlation between the vector components. There is a request for clarification and demonstration of the calculations involved. The conversation emphasizes the need for a detailed explanation and mathematical work to support the conclusion. Overall, the focus is on understanding the statistical properties of random unit vectors in three dimensions.
begyu85
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vector mean value - help!

Homework Statement



Let n = (n_1, n_2, n_3) be a random unit vector in Descartes coordinates in the 3-dimensional space.

What is the mean value (or expectation value) of n_i*n_j, where i,j = 1,2,3.

Or shortly: < n_i*n_j > = ?
 
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Pls Show Some Work.
 
i think, this mean value is a tensor of 3x3
 
Why do you think that? You still haven't shown any work.
 

Thread 'Find the polar form of a complex number'

The working out suggests first equating ## \sqrt{i} = x + iy ## and suggests that squaring and equating real and imaginary parts of both sides results in ## \sqrt{i} = \pm (1+i)/ \sqrt{2} ## Squaring both sides results in: $$ i = (x + iy)^2 $$ $$ i = x^2 + 2ixy -y^2 $$ equating real parts gives $$ x^2 - y^2 = 0 $$ $$ (x+y)(x-y) = 0 $$ $$ x = \pm y $$ equating imaginary parts gives: $$ i = 2ixy $$ $$ 2xy = 1 $$ I'm not really sure how to proceed from here.
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