Problem 46 - Calculus for Biology and Medicine

Question

Use a rotation matrix to rotate the vector \(\left[\begin{array}{r}1 \\\ -2\end{array}\right]\) clockwise by the angle \(\pi / 3\)

Step-by-Step Solution

Answer

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

Step 1: Define the Rotation Matrix

The rotation matrix for a clockwise rotation by an angle \( \theta \) is given by: \[ R = \begin{bmatrix} \cos(\theta) & \sin(\theta) \ -\sin(\theta) & \cos(\theta) \end{bmatrix} \] For a clockwise rotation by \( \pi/3 \), our \( \theta \) is \( -\pi/3 \).

Step 2: Calculate Cosine and Sine Values

Calculate \( \cos(-\pi/3) \) and \( \sin(-\pi/3) \). Using angle identities, \( \cos(-\pi/3) = \cos(\pi/3) = \frac{1}{2} \) and \( \sin(-\pi/3) = -\sin(\pi/3) = -\frac{\sqrt{3}}{2} \).

Step 3: Construct the Rotation Matrix

Using the calculated values, construct the rotation matrix \( R \): \[ R = \begin{bmatrix} \frac{1}{2} & -\frac{\sqrt{3}}{2} \ \frac{\sqrt{3}}{2} & \frac{1}{2} \end{bmatrix} \]

Step 4: Perform the Matrix Multiplication

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

Step 5: Simplify the Result

Perform the calculations in the multiplication: \[\begin{bmatrix} \frac{1}{2} + \sqrt{3} \ \frac{\sqrt{3}}{2} - 1 \end{bmatrix} = \begin{bmatrix} \frac{1}{2} + \sqrt{3} \ \frac{\sqrt{3}}{2} - 1 \end{bmatrix}\]

Step 6: Interpret the Result

The rotated vector is \( \begin{bmatrix} \frac{1}{2} + \sqrt{3} \ \frac{\sqrt{3}}{2} - 1 \end{bmatrix} \). This is the vector \( \begin{bmatrix} 1 \ -2 \end{bmatrix} \) rotated clockwise by \( \pi/3 \).

Key Concepts

Vector Rotation

When we talk about vector rotation, we are discussing the process of spinning a vector around a particular point or axis. In two dimensions, this typically involves rotating a vector within the Cartesian plane. Visualizing this can be akin to turning a line segment around a fixed point.
The concept of vector rotation is vital in fields ranging from physics to computer graphics. By rotating vectors, we can change their direction while maintaining magnitude.

• Rotation can be clockwise or counterclockwise.
• The angle of rotation is typically measured in radians.

For the vector rotation, a special mathematical tool, the rotation matrix, is employed. The rotation matrix changes the coordinates of the vector according to the desired rotation angle. But before diving into how exactly the matrix works, it's important to properly apply trigonometric identities to make accurate calculations.

Trigonometric Identities

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. Trigonometric identities are equations that hold for any angle and are helpful for simplifying the computations used in rotations.
For vector rotation, the primary trigonometric identities you need to apply involve the sine and cosine functions.

• They allow you to convert positive angles to negative ones and vice versa.
• Using the even and odd properties of cosine and sine, you find that eginlist - eginlist - t a n θ equals  s a ne θ over  c o s θ .
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