Problem 41 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding the Rotation Matrix

To rotate a vector in 2D space, we use a rotation matrix. For an angle \( \theta \), the rotation matrix is \[R(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{bmatrix}. \] In this problem, \( \theta = \pi/6 \).

Step 2: Calculate Trigonometric Values

Determine the cosine and sine for \( \theta = \pi/6 \). The values are \( \cos(\pi/6) = \sqrt{3}/2 \) and \( \sin(\pi/6) = 1/2 \). Use these values to fill the rotation matrix.

Step 3: Form the Rotation Matrix

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

Step 4: Apply the Rotation Matrix to the Vector

The original vector is \( \mathbf{v} = \begin{bmatrix} -1 \ 2 \end{bmatrix} \). Multiply the rotation matrix by the vector:\[\begin{bmatrix} \sqrt{3}/2 & -1/2 \ 1/2 & \sqrt{3}/2 \end{bmatrix} \begin{bmatrix} -1 \ 2 \end{bmatrix} = \begin{bmatrix} (-\sqrt{3}/2) + (-1) \ (-1/2) + (\sqrt{3}) \end{bmatrix}.\]

Step 5: Simplify the Result

Perform the multiplication and addition to get:\[\begin{bmatrix} (-\sqrt{3}/2) + (-1) \ (-1/2) + (\sqrt{3}) \end{bmatrix} = \begin{bmatrix} -\sqrt{3}/2 - 1 \ \sqrt{3} - 1/2 \end{bmatrix}.\] So the rotated vector is \( \begin{bmatrix} -\sqrt{3}/2 - 1 \ \sqrt{3} - 1/2 \end{bmatrix}. \)

Key Concepts

Trigonometric Values

Trigonometric values play a crucial role when dealing with any kind of rotation, especially in the context of a rotation matrix in 2D space. For an angle \( \theta \), you need the cosine and sine values to form the rotation matrix. These values are fundamental in determining how much a vector will rotate around the origin.

When \( \theta = \pi/6 \), or 30 degrees, the trigonometric values become very specific and are used often, not only in math, but in other fields. At this angle:

• \( \cos(\pi/6) = \sqrt{3}/2 \)
• \( \sin(\pi/6) = 1/2 \)

These specific values result from the geometry of a 30-60-90 triangle, which is one of the special triangles in trigonometry that you might come across. They are vital to fill in the slots of the rotation matrix which ultimately helps us in rotating the vector accurately.

2D Vector Rotation

2D vector rotation is a common operation in linear transformations used across various domains, like computer graphics and engineering. The task is to rotate a given vector by a certain angle, preserving its magnitude and changing its orientation.

To perform a 2D vector rotation, follow these steps:

• Start with a vector (for example, \( \mathbf{v} = \begin{bmatrix} -1 \ 2 \end{bmatrix} \)).
• Use a rotation matrix to rotate this vector. For a counterclockwise rotation by an angle \( \theta \), the matrix will be \[ R(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{bmatrix}. \]
• Substitute the trigonometric values into this matrix and multiply it by the vector.

The multiplication of the matrix and vector transforms the vector linearly, reorienting it to its new position without changing its size.

Linear Algebra

Linear algebra provides tools for solving systems of linear equations and transforming geometric objects. One of its applications is the manipulation of vectors and matrices.

Here's how it connects to vector rotations:

• The rotation matrix is a special matrix used to rotate vectors in a plane. It utilizes linear algebra principles, specifically transformations.
• Multiplying this matrix by a vector is an example of performing a linear transformation, where the operation can be visualized as moving a vector through a specified rotation.

Through linear algebra, we streamline complex operations and enable efficient computation of geometric transformations, such as rotations. This process allows for the modification of vector orientations without altering their lengths, which is highly useful in graphics rendering and modeling real-world scenarios.