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 \(\pi / 3\).

Step-by-Step Solution

Answer

The rotated vector is approximately \( \begin{bmatrix} 2.866 \\ 2.964 \end{bmatrix} \).

Step 1: Understand the Rotation Matrix

A rotation matrix for rotating a vector in 2D space counterclockwise by an angle \( \theta \) is given by:\[R(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{bmatrix}\]For our problem, \( \theta = \pi/3 \).

Step 2: Calculate the Values of Trigonometric Functions

For \( \theta = \pi/3 \), we calculate \( \cos(\pi/3) \) and \( \sin(\pi/3) \):\[\cos(\pi/3) = \frac{1}{2}, \quad \sin(\pi/3) = \frac{\sqrt{3}}{2}\]These values will be used to construct the rotation matrix.

Step 3: Construct the Rotation Matrix

Substitute \( \cos(\pi/3) \) and \( \sin(\pi/3) \) into the rotation matrix:\[R(\pi/3) = \begin{bmatrix} \frac{1}{2} & -\frac{\sqrt{3}}{2} \ \frac{\sqrt{3}}{2} & \frac{1}{2} \end{bmatrix}\]This matrix will rotate any 2D vector by \( \pi/3 \) radians.

Step 4: Multiply the Rotation Matrix by the Vector

Multiply the rotation matrix \( R(\pi/3) \) by the vector \( \begin{bmatrix} 4 \ -1 \end{bmatrix} \):\[\begin{bmatrix} \frac{1}{2} & -\frac{\sqrt{3}}{2} \ \frac{\sqrt{3}}{2} & \frac{1}{2} \end{bmatrix} \begin{bmatrix} 4 \ -1 \end{bmatrix} = \begin{bmatrix} \frac{1}{2} \times 4 + \left(-\frac{\sqrt{3}}{2}\right) \times (-1) \ \frac{\sqrt{3}}{2} \times 4 + \frac{1}{2} \times (-1) \end{bmatrix}\]Calculate the result for both entries.

Step 5: Perform the Matrix Multiplication

Compute each element of the resulting vector:\[\begin{align*}x' &= \frac{1}{2} \times 4 + (-\frac{\sqrt{3}}{2}) \times (-1) = 2 + \frac{\sqrt{3}}{2} = 2 + 0.866 \approx 2.866,\y' &= \frac{\sqrt{3}}{2} \times 4 + \frac{1}{2} \times (-1) = 2\sqrt{3} - \frac{1}{2} = 3.464 - 0.5 \approx 2.964.\end{align*}\]So, the rotated vector is \( \begin{bmatrix} 2.866 \ 2.964 \end{bmatrix} \).

Key Concepts

Trigonometric Functions

Trigonometric functions play a crucial role in understanding rotations in mathematics, especially when dealing with vectors. They help us describe how angles and distances relate on a unit circle. When rotating vectors, the two primary trigonometric functions we use are cosine and sine. These functions provide the components needed to create a rotation matrix.

• Cosine Function: In a right triangle, the cosine of an angle \( \theta \) is the ratio of the length of the adjacent side to the hypotenuse. Cosine values range from -1 to 1 and are periodic with period \( 2\pi \).
• Sine Function: Similarly, the sine of an angle \( \theta \) is the ratio of the length of the opposite side to the hypotenuse. Sine values also range from -1 to 1, having the same periodic nature as cosine.

For the angle \( \theta = \pi/3 \), which is equivalent to 60 degrees, the cosine and sine values are particularly memorable: \( \cos(\pi/3) = 1/2 \) and \( \sin(\pi/3) = \sqrt{3}/2 \). These values are essential for constructing the rotation matrix, aiding in vector transformations across a plane.

Vector Rotation

Vector rotation is a method of altering the direction of a vector while maintaining its magnitude. In 2D, this process uses the rotation matrix. This matrix acts like a tool to pivot vectors around the origin, depending on the specified angle, noted as \( \theta \).
The 2D rotation matrix for counterclockwise rotation is given by:

• \( \begin{bmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{bmatrix} \)

This matrix effectively changes the original vector's orientation. By using trigonometric identities, like those calculated for \( \pi/3 \) in this case, we create a rotation matrix:

• \( \begin{bmatrix} \frac{1}{2} & -\frac{\sqrt{3}}{2} \ \frac{\sqrt{3}}{2} & \frac{1}{2} \end{bmatrix} \)

By multiplying this matrix by a vector, the components appropriately realign, resulting in a transformed vector with the same magnitude. This technique ensures consistent and predictable adjustments to vector directions based on rotational angles.

2D Vectors

2D vectors are fundamental elements in mathematics and physics, representing quantities with both magnitude and direction in a two-dimensional plane. Understanding vectors is essential for numerous practical applications, like physics simulations and computer graphics.
A 2D vector can be denoted in the form \( \begin{bmatrix} x \ y \end{bmatrix} \), where \( x \) and \( y \) are the components along the horizontal and vertical axes respectively.
When it comes to rotation, applying mathematical operations like matrix multiplication helps in manipulating these vectors. Let's consider a 2D vector \( \begin{bmatrix} 4 \ -1 \end{bmatrix} \) from the problem. By executing matrix multiplication with the rotation matrix derived, we effectively alter its direction from the original orientation to a new one by the angle \( \pi/3 \). The new vector becomes \( \begin{bmatrix} 2.866 \ 2.964 \end{bmatrix} \). This transformation maintains the original vector's magnitude while modifying its direction within the 2D plane. By mastering the manipulation of 2D vectors, one gains insights into the dynamics of movement and the effects of directional changes in various scientific fields.