Problem 44 - Calculus for Biology and Medicine

Question

Use a rotation matrix to rotate the vector \(\left[\begin{array}{l}-2 \\\ -3\end{array}\right]\) counterclockwise by the angle \(\pi / 9\).

Step-by-Step Solution

Answer

The rotated vector is \( \begin{bmatrix} -0.8534 \\ -3.5031 \end{bmatrix} \).

Step 1: Understand the Rotation Matrix

A rotation matrix for a counterclockwise rotation by angle \( \theta \) is given by:\[R = \begin{bmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{bmatrix}\]For this problem, we need to substitute \( \theta = \frac{\pi}{9} \) into the matrix.

Step 2: Calculate Trigonometric Values

First, calculate \( \cos \frac{\pi}{9} \) and \( \sin \frac{\pi}{9} \). Using trigonometric tables or a calculator, we have:\[ \cos \frac{\pi}{9} \approx 0.9397, \quad \sin \frac{\pi}{9} \approx 0.3420 \]

Step 3: Form the Rotation Matrix

Substitute the trigonometric values into the rotation matrix formula to form:\[R = \begin{bmatrix} 0.9397 & -0.3420 \ 0.3420 & 0.9397 \end{bmatrix}\]

Step 4: Multiply the Rotation Matrix by the Vector

We need to multiply the rotation matrix by the given vector \( \mathbf{v} = \begin{bmatrix} -2 \ -3 \end{bmatrix} \). This gives:\[\mathbf{v}' = \begin{bmatrix} 0.9397 & -0.3420 \ 0.3420 & 0.9397 \end{bmatrix} \begin{bmatrix} -2 \ -3 \end{bmatrix}\]

Step 5: Perform the Matrix Multiplication

Perform the matrix multiplication:\[\mathbf{v}' = \begin{bmatrix} 0.9397 \times (-2) + (-0.3420) \times (-3) \ 0.3420 \times (-2) + 0.9397 \times (-3) \end{bmatrix}\]Calculate each component:\[\mathbf{v}' = \begin{bmatrix} -1.8794 + 1.0260 \ -0.6840 - 2.8191 \end{bmatrix} = \begin{bmatrix} -0.8534 \ -3.5031 \end{bmatrix}\]

Step 6: Write the Final Result

The rotated vector is \( \begin{bmatrix} -0.8534 \ -3.5031 \end{bmatrix} \). This is the vector \( \mathbf{v} \) after rotation by an angle of \( \frac{\pi}{9} \) counterclockwise.

Key Concepts

Trigonometric Functions

Trigonometric functions are essential tools in geometry, primarily dealing with angles and sides of triangles. In this section, we specifically focus on sine and cosine functions, which are pivotal for forming rotation matrices.

• Sine ( \( \sin \theta \)): Represents the vertical component of an angle in a unit circle.
• Cosine ( \( \cos \theta \)): Indicates the horizontal component.

These functions appear in the rotation matrix formulation used to rotate vectors in a plane. For a rotation matrix transforming a vector counterclockwise by an angle \( \theta \), they are placed as \(\[R = \begin{bmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{bmatrix}\]\).

In our problem, the angle is \( \pi/9 \), making the components \( \cos(\pi/9) \approx 0.9397 \) and \( \sin(\pi/9) \approx 0.3420 \). These trigonometric values form the basis for calculating how the vector will rotate around the origin.

Matrix Multiplication

Matrix multiplication extends the concept of multiplication beyond numbers, applying it to matrices that consist of rows and columns of numbers. In vector rotation, matrix multiplication works to transform the vector through the rotational transformation specified by the rotation matrix.

Here's a brief breakdown of how it works:

• Identify each row of the rotation matrix.
• Multiply corresponding components of the matrix row and vector column together.
• Sum these products to find each element of the resulting vector.

For the vector rotation example, you multiply the rotation matrix \(\[ \begin{bmatrix} 0.9397 & -0.3420 \ 0.3420 & 0.9397 \end{bmatrix} \]\) with the vector \(\[ \begin{bmatrix} -2 \ -3 \end{bmatrix} \]\): - Calculate top component: \(0.9397 \times (-2) + (-0.3420) \times (-3)\) - For the bottom component: \(0.3420 \times (-2) + 0.9397 \times (-3)\)These products determine the new position of the rotated vector in the coordinate plane.

Vector Transformation

A vector transformation alters a vector's position in a plane or space, maintaining its magnitude or direction depending on the transformation. In context, we use a rotation matrix to perform this transformation to pivot vectors around the origin.

The transformation using a rotation matrix involves recalculating the vector's components:

• Initial vector: \(\[ \begin{bmatrix} -2 \ -3 \end{bmatrix} \]\)
• Applied Rotation Matrix: \(\[ \begin{bmatrix} 0.9397 & -0.3420 \ 0.3420 & 0.9397 \end{bmatrix} \]\)

This process yields a new vector representing the old vector's position rotated by \( \frac{\pi}{9} \) radians counterclockwise. The outcome, \(\[ \begin{bmatrix} -0.8534 \ -3.5031 \end{bmatrix} \]\), illustrates how mathematical operations alter a vector's orientation while conserving its initial characteristics under a specified rotation.