Answered: The matrix P that multiplies ( x, y, z)…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

The matrix P that multiplies ( x, y, z) to give ( z, x, y) is also a rotation matrix.
Find P and P3. The rotation axis a = (l, 1, 1) doesn't move, it equals Pa. What is the angle of rotation from v = (2, 3, -5) to Pv = (-5, 2, 3)?

Expert Solution

Step 1

We have to find the matrix P such that $\begin{array}{rcl} P (\begin{matrix} x \\ y \\ z \end{matrix}) & = & (\begin{matrix} z \\ x \\ y \end{matrix}) \end{array}$ where P is the rotation matrix. Then, we have to

find the matrix $P^{3}$ .

We know that a rotation matrix is a square matrix whose all entries are real numbers. The determinant

of a rotation matrix is equal to 1.

Step 2

Now,

$\begin{array}{rcl} P (\begin{matrix} x \\ y \\ z \end{matrix}) & = & (\begin{matrix} z \\ x \\ y \end{matrix}) \\ (\begin{matrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}) (\begin{matrix} x \\ y \\ z \end{matrix}) & = & (\begin{matrix} z \\ x \\ y \end{matrix}) \\ (\begin{matrix} 0 \cdot x + 0 \cdot y + 1 \cdot z \\ 1 \cdot x + 0 \cdot y + 0 \cdot z \\ 0 \cdot x + 1 \cdot y + 0 \cdot z \end{matrix}) & = & (\begin{matrix} z \\ x \\ y \end{matrix}) \\ (\begin{matrix} z \\ x \\ y \end{matrix}) & = & (\begin{matrix} z \\ x \\ y \end{matrix}) \end{array}$

Therefore, $\begin{array}{rcl} P & = & (\begin{matrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}) \end{array}$ .

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