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Chapter 1: Problem 20

The matrix that rotates the \(x-y\) plane by an angle \(\theta\) is $$ A(\theta)=\left[\begin{array}{rr} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right] $$ Verify that \(A\left(\theta_{1}\right) A\left(\theta_{2}\right)=A\left(\theta_{1}+\theta_{2}\right)\) from the identities for \(\cos \left(\theta_{1}+\theta_{2}\right)\) and \(\sin \left(\theta_{1}+\theta_{2}\right)\). What is \(A(\theta)\) times \(A(-\theta)\) ?

Short Answer

Expert verified
\(A(\theta_1)A(\theta_2) = A(\theta_1+\theta_2)\); \(A(\theta)A(-\theta)\) is the identity matrix.

Step by step solution

01

Define the Rotation Matrices

Define the rotation matrices \(A(\theta_1)\) and \(A(\theta_2)\) as given in the problem:\[A(\theta_1) = \begin{bmatrix}\cos \theta_1 & -\sin \theta_1 \\sin \theta_1 & \cos \theta_1\end{bmatrix}\]\[A(\theta_2) = \begin{bmatrix}\cos \theta_2 & -\sin \theta_2 \\sin \theta_2 & \cos \theta_2\end{bmatrix}\]
02

Calculate the Product of Two Rotation Matrices

Compute the product \(A(\theta_1)A(\theta_2)\):\[A(\theta_1) A(\theta_2) = \begin{bmatrix}\cos \theta_1 & -\sin \theta_1 \\sin \theta_1 & \cos \theta_1\end{bmatrix} \begin{bmatrix}\cos \theta_2 & -\sin \theta_2 \\sin \theta_2 & \cos \theta_2\end{bmatrix}\]Perform matrix multiplication:- Top-left element: \(\cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2\)- Top-right element: \(-\cos \theta_1 \sin \theta_2 - \sin \theta_1 \cos \theta_2\)- Bottom-left element: \(\sin \theta_1 \cos \theta_2 + \cos \theta_1 \sin \theta_2\)- Bottom-right element: \(-\sin \theta_1 \sin \theta_2 + \cos \theta_1 \cos \theta_2\)So,\[A(\theta_1) A(\theta_2) = \begin{bmatrix}\cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2 & -\cos \theta_1 \sin \theta_2 - \sin \theta_1 \cos \theta_2 \\sin \theta_1 \cos \theta_2 + \cos \theta_1 \sin \theta_2 & \cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2\end{bmatrix}\]
03

Use Trigonometric Identities

Apply the cosine and sine addition formulas:\[\cos(\theta_1 + \theta_2) = \cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2\]\[\sin(\theta_1 + \theta_2) = \sin \theta_1 \cos \theta_2 + \cos \theta_1 \sin \theta_2\]Substitute these formulas into the matrix:\[A(\theta_1 + \theta_2) = \begin{bmatrix}\cos(\theta_1 + \theta_2) & -\sin(\theta_1 + \theta_2) \\sin(\theta_1 + \theta_2) & \cos(\theta_1 + \theta_2)\end{bmatrix}\]
04

Compare and Confirm Matrix Equality

The expression from Step 2:\[\begin{bmatrix}\cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2 & -\cos \theta_1 \sin \theta_2 - \sin \theta_1 \cos \theta_2 \\sin \theta_1 \cos \theta_2 + \cos \theta_1 \sin \theta_2 & \cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2\end{bmatrix}\]matches the form:\[A(\theta_1 + \theta_2) = \begin{bmatrix}\cos(\theta_1 + \theta_2) & -\sin(\theta_1 + \theta_2) \\sin(\theta_1 + \theta_2) & \cos(\theta_1 + \theta_2)\end{bmatrix}\]Thus, we confirm that\(A(\theta_1)A(\theta_2) = A(\theta_1 + \theta_2)\).
05

Calculate \(A(\theta)A(-\theta)\)

Compute the matrix \(A(-\theta)\):\[A(-\theta) = \begin{bmatrix}\cos(-\theta) & -\sin(-\theta) \\sin(-\theta) & \cos(-\theta)\end{bmatrix} = \begin{bmatrix}\cos \theta & \sin \theta \-\sin \theta & \cos \theta\end{bmatrix}\]Calculate the product \(A(\theta) \cdot A(-\theta)\):\[\begin{bmatrix}\cos \theta & -\sin \theta \\sin \theta & \cos \theta\end{bmatrix} \begin{bmatrix}\cos \theta & \sin \theta \-\sin \theta & \cos \theta\end{bmatrix} = \begin{bmatrix}\cos^2 \theta + \sin^2 \theta & 0 \0 & \cos^2 \theta + \sin^2 \theta\end{bmatrix} = \begin{bmatrix}1 & 0 \0 & 1\end{bmatrix}\]Thus, \(A(\theta)A(-\theta)\) equals the identity matrix.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Multiplication
Matrix multiplication is a method used to combine two matrices into a new matrix. This operation is especially important in linear transformations, including rotation matrices. When multiplying two matrices, the order matters because matrix multiplication is not commutative. This means that in general, \( AB eq BA \).To multiply matrices, follow these steps:
  • Take the elements of the first row of the first matrix.
  • Pair them with the corresponding elements of the first column of the second matrix.
  • Multiply these pairs and sum them to get the new element in the resulting matrix.
  • Repeat this process for each row and column.
In the context of rotation matrices, matrix multiplication allows us to combine rotations efficiently. For instance, if two rotation matrices are multiplied, the result is a new rotation matrix equivalent to a single rotation by the sum of the two angles. This property is a consequence of how trigonometric functions work in the rotation matrices.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values of the included variables. They are essential tools in simplifying expressions and solving equations involving trigonometric functions.Two key identities often used with rotation matrices are:
  • Cosine addition formula: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \)
  • Sine addition formula: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)
In rotation matrices, these identities enable simplification of the calculated product of two matrices into another rotation matrix. Specifically, when we calculate \( A(\theta_1)A(\theta_2) \) for rotation matrices, these identities help us match the result to the rotation matrix of angle \( \theta_1 + \theta_2 \). Thus, the compositions of rotations are straightforward because of these identities.
Identity Matrix
The identity matrix plays a critical role in matrix operations as it behaves similarly to the number 1 in multiplication for regular numbers. It is a square matrix with ones on the diagonal and zeros elsewhere. In 2x2 matrices, it looks like:\[I = \begin{bmatrix}1 & 0 \0 & 1\end{bmatrix}\]When any matrix \( A \) is multiplied by the identity matrix \( I \), the result is the matrix \( A \) itself. This confirms that multiplying by an identity matrix doesn't alter the original matrix:- Matrix \( A \) after multiplication: \( AI = IA = A \)In the context of rotation matrices, when you multiply a rotation matrix \( A(\theta) \) by \( A(-\theta) \), the result is the identity matrix. This essentially shows how a rotation followed by its reverse cancels out, leaving no change in the plane. Hence, the identity matrix can be seen as a representation of no rotation or zero-degree rotation in the rotation of planes.

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Most popular questions from this chapter

Starting from a 3 by 3 matrix \(A\) with pivots \(2,7,6\), add a fourth row and column to produce \(M\). What are the first three pivots for \(M\), and why? What fourth row and column are sure to produce 9 as the fourth pivot?

Invent a 3 by 3 magic matrix \(M\) with entries \(1,2, \ldots, 9\). All rows and columns and diagonals add to 15 . The first row could be \(8,3,4\). What is \(M\) times \((1,1,1)\) ? What is the row vector \(\left.\begin{array}{lll}1 & 1 & 1\end{array}\right]\) times \(M ?\)

Write \(2 x+3 y+z+5 t=8\) as a matrix \(A\) (how many rows?) multiplying the column vector \((x, y, z, t)\) to produce \(b\). The solutions fill a plane in four-dimensional space. The plane is three-dimensional with no \(4 \mathrm{D}\) volume.

Find the inverses (no special system required) of $$ A_{1}=\left[\begin{array}{ll} 0 & 2 \\ 3 & 0 \end{array}\right], \quad A_{2}=\left[\begin{array}{ll} 2 & 0 \\ 4 & 2 \end{array}\right], \quad A_{3}=\left[\begin{array}{rr} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right] $$

With \(h=\frac{1}{4}\) and \(f(x)=4 \pi^{2} \sin 2 \pi x\), the difference equation (5) is $$ \left[\begin{array}{rrr} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{array}\right]\left[\begin{array}{l} u_{1} \\ u_{2} \\ u_{3} \end{array}\right]=\frac{\pi^{2}}{4}\left[\begin{array}{r} 1 \\ 0 \\ -1 \end{array}\right] $$ Solve for \(u_{1}, u_{2}, u_{3}\) and find their error in comparison with the true solution \(u=\) \(\sin 2 \pi x\) at \(x=\frac{1}{4}, x=\frac{1}{2}\), and \(x=\frac{3}{4}\).
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