Problem 42 - Calculus for Biology and Medicine

Question

Use a rotation matrix to rotate the vector \(\left[\begin{array}{r}4 \\\ -1\end{array}\right]\) counterclockwise by the angle \(\pi / 6\).

Step-by-Step Solution

Answer

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

Step 1: Write Down the Vector and Angle

The vector we want to rotate is \( \begin{bmatrix} 4 \ -1 \end{bmatrix} \) and the angle of rotation is \( \pi / 6 \).

Step 2: Define the Rotation Matrix

The rotation matrix for a counterclockwise rotation by an angle \( \theta \) is given by \[ R = \begin{bmatrix} \cos\theta & -\sin\theta \ \sin\theta & \cos\theta \end{bmatrix} \]. For \( \theta = \pi / 6 \), its cosine and sine are \( \cos(\pi / 6) = \frac{\sqrt{3}}{2} \) and \( \sin(\pi / 6) = \frac{1}{2} \).

Step 3: Construct the Rotation Matrix

Insert the values from Step 2 into the rotation matrix: \[ R = \begin{bmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} \ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{bmatrix} \].

Step 4: Perform Matrix Multiplication

Multiply the rotation matrix by the vector: \[ \begin{bmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} \ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{bmatrix} \begin{bmatrix} 4 \ -1 \end{bmatrix} = \begin{bmatrix} \frac{\sqrt{3}}{2}(4) + \frac{1}{2}(1) \ \frac{1}{2}(4) + \frac{\sqrt{3}}{2}(-1) \end{bmatrix} \].

Step 5: Calculate the New Vector Components

Calculate each component separately: - For the first component: \( 4 \times \frac{\sqrt{3}}{2} + 1 \times \frac{1}{2} = 2\sqrt{3} + \frac{1}{2} \).- For the second component: \( 4 \times \frac{1}{2} - 1 \times \frac{\sqrt{3}}{2} = 2 - \frac{\sqrt{3}}{2} \).

Step 6: Write the Resulting Rotated Vector

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

Key Concepts

Matrix Multiplication

Matrix multiplication is an essential operation in linear algebra, often used to transform vectors in space. To multiply two matrices, we take each row of the first matrix and each column of the second matrix, multiplying their corresponding elements and summing the results. This process produces a new matrix.

• The number of columns in the first matrix must be equal to the number of rows in the second matrix to perform the multiplication.
• In this problem, we are multiplying a 2x2 matrix (the rotation matrix) with a 2x1 matrix (the vector).
• This operation results in a new 2x1 vector, representing the transformed coordinates of the original vector after rotation.

Understanding matrix multiplication is crucial because it allows us to apply complex transformations, such as rotations, translations, and scalings, in a clear and consistent manner.

Vector Rotation

Vector rotation involves changing the direction of a vector while maintaining its magnitude. It is achieved using a rotation matrix, a very efficient tool in 2D and 3D geometry. For 2D vector rotation:

• The rotation matrix is \[ R = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{bmatrix} \]
• By multiplying the rotation matrix with the vector you wish to rotate, you receive the rotated vector.
• This operation shifts the vector counterclockwise or clockwise, depending on the angle.

In this problem, the vector \( \begin{bmatrix} 4 \ -1 \end{bmatrix} \)is rotated by an angle \( \pi / 6 \) radians counterclockwise.This transformation causes the vector to 'swing' in the geometric space, moving it to a new position but keeping its initial size.

Trigonometric Functions

Trigonometric functions are vital in calculating rotation matrices and understanding the transformations they produce. When dealing with rotations, cosines and sines are the primary functions used:

• Cosine, denoted as \(\cos(\theta)\), helps determine the horizontal displacement during the rotation.
• Sine, denoted as \(\sin(\theta)\), contributes to the vertical displacement.
• In this exercise, we employ the values \(\cos(\pi/6) = \frac{\sqrt{3}}{2}\) and \(\sin(\pi/6) = \frac{1}{2}\).

Using these trigonometric functions in the rotation matrix, you can calculate the precise new coordinates of a vector after a rotation. Understanding the role of these functions not only aids in manual calculations but also in appreciating how rotations are deducted through function values.