Question

Rotate the given quadric surface to principal axes. What is the name of the surface? What is the shortest distance from the origin to the surface? $$x^{2}+y^{2}-5 z^{2}+4 x y=15$$

Short answer

The surface is a hyperboloid of one sheet. The shortest distance from the origin to the surface is \( \sqrt{5} \).

Step-by-step solution

Identify the quadric surface

The given equation is: \[ x^{2} + y^{2} - 5z^{2} + 4xy = 15 \] This represents a quadric surface in three-dimensional space.

Write the quadratic form

Rewriting the given quadric surface using the quadratic form matrix representation: \[ x Q x^{T} = 15 \] where \[ Q = \begin{bmatrix} 1 & 2 & 0 \ 2 & 1 & 0 \ 0 & 0 & -5 \end{bmatrix} \] and \[ x = \begin{bmatrix} x \ y \ z \end{bmatrix} \]

Determine the eigenvalues and eigenvectors

To rotate the quadric surface to principal axes, find the eigenvalues and eigenvectors of matrix Q: Solving the characteristic equation \( \det(Q - \lambda I) = 0 \) gives the eigenvalues \( \lambda_1 = -1, \lambda_2 = 3, \lambda_3 = -5 \). The corresponding eigenvectors are:

Diagonalize the matrix Q

Using the eigenvalues and eigenvectors, form the diagonal matrix and the transformation matrix P: \[ P = \text{[Eigenvectors]} \] \[ P^{-1} Q P = \begin{bmatrix} -1 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & -5 \end{bmatrix} \]

Identify the rotated equation

Transform the equation using the diagonal matrix: \[ -u^{2} + 3v^{2} - 5w^{2} = 15 \]

Identify the name of the surface

The final form of the equation \( -u^{2} + 3v^{2} - 5w^{2} = 15 \) indicates that the surface is a hyperboloid of one sheet.

Determine the shortest distance from the origin

To find the shortest distance from the origin to the hyperboloid: Set u = 0, v = 0, and solve for w in \( -5w^{2} = 15 \), giving \( w^2 = -3 \). This doesn't yield a real solution. Set u = 0, w = 0, and solve for v in \( 3v^{2} = 15 \), giving \( v = \sqrt{5} \). Thus, the shortest distance is \( \sqrt{15/3} = \sqrt{5} \). So, the shortest distance is \( \sqrt{5} \).

Key concepts

Eigenvalues and Eigenvectors

Understanding eigenvalues and eigenvectors is key to solving many problems in linear algebra, especially those involving matrices. Eigenvalues are scalars that provide important insights into the properties of a matrix. When you have a matrix \( A \), an eigenvalue \( \lambda \) is a solution to the characteristic equation \( \det(A - \lambda I) = 0 \). Here, \( I \) is the identity matrix.

Eigenvectors are non-zero vectors that, when multiplied by the matrix, result in a scaled version of the original vector, defined mathematically as \( Av = \lambda v \), where \( v \) is the eigenvector and \( \lambda \) is the eigenvalue.

In the given problem, the matrix \( Q \) has eigenvalues \( \lambda_1 = -1, \lambda_2 = 3, \lambda_3 = -5 \). These values are derived by solving the characteristic equation for \( Q \). The eigenvalues reflect how the quadratic form is stretched or compressed along certain directions, which are defined by their corresponding eigenvectors.

Diagonalization of Matrices

When we talk about diagonalizing a matrix, we mean converting it into a diagonal matrix through a similarity transformation. Diagonalization simplifies many computations, including those involving powers of matrices and solving related systems of equations.

A matrix \( A \) is diagonalizable if there exists a matrix \( P \) such that \( P^{-1}AP \) is a diagonal matrix. The columns of \( P \) are the eigenvectors of \( A \), and the diagonal entries of \( P^{-1}AP \) are the eigenvalues of \( A \).

In our problem, matrix \( Q \) is diagonalized to form \( P^{-1} Q P = \begin{bmatrix} -1 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & -5 \end{bmatrix} \). This transformation simplifies the original quadratic form, effectively rotating the coordinate system to align with the principal axes defined by the eigenvectors.

Hyperboloid

A hyperboloid is a type of quadric surface that can be either one-sheeted or two-sheeted, depending on the signs in its equation. For instance, a hyperboloid of one sheet has the general form \( -u^2 + v^2 + w^2 = 1 \), while a hyperboloid of two sheets might be expressed as \( -u^2 + v^2 - w^2 = 1 \).

In the given problem, the equation \( -u^2 + 3v^2 - 5w^2 = 15 \) identifies a hyperboloid of one sheet. These surfaces are characterized by their saddle-like structure and their ability to extend infinitely.

The hyperboloid is useful in various scientific fields, including physics and engineering, because of its geometric properties. For instance, hyperboloids can be found in cooling towers of power plants and in various architectural structures.

To find the shortest distance from the origin to this hyperboloid, you suitably adjust the parameters to emphasize the coordinate contributing the least to the equation, ending up with the distance \( \sqrt{5} \) from the origin in this case.