Problem 46 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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, \(\theta = \pi/6\).

Step 2: Calculate the Sine and Cosine

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

Step 3: Construct the Rotation Matrix

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

Step 4: Multiply the Rotation Matrix by the Vector

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

Step 5: 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:\[\begin{bmatrix} \sqrt{3}/2 - 1 \ 1/2 + \sqrt{3} \end{bmatrix}\]

Key Concepts

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’s 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’s position through rotation or dilation. Rotation rotates the vector around the origin by a given angle. Here’s 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.