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Chapter 9: Problem 46

Use a rotation matrix to rotate the vector \(\left[\begin{array}{l}1 \\\ 2\end{array}\right]\) counterclockwise by the angle \(\pi / 6\).

Short Answer

Expert verified
The rotated vector is \(\begin{bmatrix} \sqrt{3}/2 - 1 \\ 1/2 + \sqrt{3} \end{bmatrix}\).

Step by step solution

01

Understand the Rotation Matrix

A rotation matrix for counterclockwise rotation by an angle \(\theta\) is given by the matrix: \[R = \begin{bmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{bmatrix}\]. For this exercise, \(\theta = \pi/6\).
02

Calculate the Sine and Cosine

To fill in the rotation matrix, we need \(\cos(\pi/6)\) and \(\sin(\pi/6)\). First, calculate these values:- \(\cos(\pi/6) = \sqrt{3}/2\)- \(\sin(\pi/6) = 1/2\)
03

Construct the Rotation Matrix

Substitute the values of \(\cos(\pi/6)\) and \(\sin(\pi/6)\) into the rotation matrix:\[R = \begin{bmatrix} \sqrt{3}/2 & -1/2 \ 1/2 & \sqrt{3}/2 \end{bmatrix}\]
04

Multiply the Rotation Matrix by the Vector

Now, multiply the rotation matrix \(R\) by the vector \(\begin{bmatrix} 1 \ 2 \end{bmatrix}\):\[\begin{bmatrix} \sqrt{3}/2 & -1/2 \ 1/2 & \sqrt{3}/2 \end{bmatrix} \cdot \begin{bmatrix} 1 \ 2 \end{bmatrix} = \begin{bmatrix} (\sqrt{3}/2) \cdot 1 + (-1/2) \cdot 2 \ (1/2) \cdot 1 + (\sqrt{3}/2) \cdot 2 \end{bmatrix}\]
05

Simplify the Resulting Vector

Carry out the multiplications and additions:- For the first component: \((\sqrt{3}/2) \cdot 1 + (-1/2) \cdot 2 = \sqrt{3}/2 - 1\)- For the second component: \((1/2) \cdot 1 + (\sqrt{3}/2) \cdot 2 = 1/2 + \sqrt{3}\).Thus, the rotated vector is:\[\begin{bmatrix} \sqrt{3}/2 - 1 \ 1/2 + \sqrt{3} \end{bmatrix}\]

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Key Concepts

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

Trigonometric Functions
Trigonometric functions, such as sine (\(\sin\theta\)) and cosine (\(\cos\theta\)), play a crucial role in understanding rotations. They help us describe how much a vector or point should rotate around an origin. Trigonometric functions are defined on the unit circle, where each angle corresponds to a point on the circle. Here鈥檚 how these functions are applied:
  • **Cosine**: Represents the horizontal component of an angle in a rotation. For \(\theta = \pi/6\), \(\cos(\pi/6) = \sqrt{3}/2\).
  • **Sine**: Represents the vertical component of an angle in a rotation. For \(\theta = \pi/6\), \(\sin(\pi/6) = 1/2\).
Understanding these functions is essential when constructing and applying the rotation matrix to perform transformations.
Vector Transformation
Vector transformation is key when dealing with rotation matrices. Essentially, it involves changing the vector鈥檚 position through rotation or dilation. Rotation rotates the vector around the origin by a given angle. Here鈥檚 how it works:
  • **Initial Vector**: In our example, the initial vector is \(\begin{bmatrix} 1 \ 2 \end{bmatrix}\).
  • **Rotation Process**: Calculate new components as a result of the rotation. This involves using trigonometric functions from the rotation matrix matching specific angles.
When you apply the rotation, you transform the vector into a new position, maintaining its magnitude but altering its direction.
Matrix Multiplication
Matrix multiplication is a fundamental operation in linear algebra used to rotate vectors. Understanding how it works is crucial for vector transformation.
  • **Matrix Setup**: Given the rotation matrix \(R = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{bmatrix}\) and the vector \(\begin{bmatrix} 1 \ 2 \end{bmatrix}\), organize them correctly for multiplication.
  • **Perform Multiplication**: Use the formula \(R \cdot \vec{v}\) by multiplying rows of the matrix by columns of the vector:
    • For the first component, compute \((\sqrt{3}/2) \cdot 1 + (-1/2) \cdot 2\).
    • For the second component, compute \((1/2) \cdot 1 + (\sqrt{3}/2) \cdot 2\).
  • **Result**: The final step gives you the new vector after rotation \(\begin{bmatrix} \sqrt{3}/2 - 1 \ 1/2 + \sqrt{3} \end{bmatrix}\).
Through systematic multiplication of matrices and vectors, you can easily perform complex transformations like rotations.

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Most popular questions from this chapter

Let \(A=(-1,0)\) and \(B=(2,-3)\). Find the vector representation of \(\overrightarrow{A B}\).

Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) for $$ A=\left[\begin{array}{ll} a & c \\ 0 & b \end{array}\right] $$

Find the parametric equation of the line in \(x-y-z\) space that goes through the indicated point in the direction of the indicated vector. $$(2,1,-3),\left[\begin{array}{r}3 \\ -1 \\ 2\end{array}\right]$$

Let \(\mathbf{x}=[3,-2,1]^{\prime}\). Find any vector \(\mathbf{y}\) so that \(\mathbf{x}\) and \(\mathbf{y}\) are perpendicular. [Your solution will not be unique.]

Find the equation of the plane through \((0,0,0)\) and perpendicular to \([1,1,1]\) '.
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