**Explanation:**

To understand why the value of the dot product of the eigenvectors corresponding to any pair of different eigenvalues of a 4-by-4 symmetric positive definite matrix is 0, let's break down the concepts involved.

**Eigenvectors and Eigenvalues:**

In linear algebra, an eigenvector of a square matrix is a nonzero vector that, when multiplied by the matrix, only changes by a scalar factor. The scalar factor is called the eigenvalue corresponding to that eigenvector.

For any given matrix A, an eigenvector x and its corresponding eigenvalue λ satisfy the equation Ax = λx.

**Symmetric Positive Definite Matrix:**

A matrix A is symmetric if it equals its own transpose, i.e., A = A^T. A positive definite matrix is a symmetric matrix in which all eigenvalues are positive.

**Dot Product of Eigenvectors:**

The dot product of two vectors u and v is calculated by multiplying their corresponding components and summing the results. Mathematically, the dot product of u and v is given by u · v = u1v1 + u2v2 + ... + unvn.

**Proof:**

1. Let A be a 4-by-4 symmetric positive definite matrix.
2. As A is symmetric, it has four distinct eigenvalues (λ1, λ2, λ3, λ4) and corresponding eigenvectors (x1, x2, x3, x4).
3. For any pair of different eigenvalues (let's consider λ1 and λ2), we need to show that the dot product of their corresponding eigenvectors (let's consider x1 and x2) is 0.
4. By definition, Ax1 = λ1x1 and Ax2 = λ2x2.
5. Taking the dot product of both sides of these equations with x1, we get x1 · Ax1 = x1 · (λ1x1) and x1 · Ax2 = x1 · (λ2x2).
6. Simplifying these equations, we have λ1(x1 · x1) = λ1||x1||^2 and λ2(x1 · x2) = λ2||x1||^2.
7. Since λ1 and λ2 are distinct eigenvalues, λ1 ≠ λ2. Therefore, we can divide both sides of the second equation by (λ1 - λ2).
8. Dividing, we get (λ1 - λ2)(x1 · x2) = 0.
9. Since λ1 - λ2 ≠ 0, the only way for the equation to hold true is if (x1 · x2) = 0.
10. Therefore, the value of the dot product of the eigenvectors corresponding to any pair of different eigenvalues of a 4-by-4 symmetric positive definite matrix is indeed 0.

Hence, the correct answer is 0.

Orthogonal,then dot have zero