Problem 47 - Calculus for Biology and Medicine

Question

Use a rotation matrix to rotate the vector \(\left[\begin{array}{r}5 \\\ -3\end{array}\right]\) clockwise by the angle \(\pi / 2\).

Step-by-Step Solution

Answer

The vector rotated clockwise by \( \pi/2 \) is \( \begin{bmatrix} 3 \\ 5 \end{bmatrix} \).

Step 1: Identify the Rotation Matrix

When rotating a 2D vector by an angle \theta clockwise, we use the following rotation matrix: \[ R = \begin{bmatrix} \cos(\theta) & \sin(\theta) \ -\sin(\theta) & \cos(\theta) \end{bmatrix} \]. Since we are rotating by \( \pi/2 \) clockwise, \( \theta = -\pi/2 \). Substitute \( \theta \) into the rotation matrix: \[ R = \begin{bmatrix} \cos(-\pi/2) & \sin(-\pi/2) \ -\sin(-\pi/2) & \cos(-\pi/2) \end{bmatrix} \].

Step 2: Compute the Cosine and Sine Values

Recall that \( \cos(-\pi/2) = 0 \) and \( \sin(-\pi/2) = -1 \). Substitute these values into the matrix: \[ R = \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} \].

Step 3: Multiply the Rotation Matrix by the Vector

We multiply the rotation matrix \( R \) by the vector \( \mathbf{v} = \begin{bmatrix} 5 \ -3 \end{bmatrix} \). Perform the matrix multiplication: \[ R\mathbf{v} = \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} \begin{bmatrix} 5 \ -3 \end{bmatrix} = \begin{bmatrix} (0)(5) + (-1)(-3) \ (1)(5) + (0)(-3) \end{bmatrix} = \begin{bmatrix} 3 \ 5 \end{bmatrix} \].

Step 4: Interpret the Result

The result of the multiplication gives the coordinates of the rotated vector. Hence, the vector \( \begin{bmatrix} 5 \ -3 \end{bmatrix} \) rotated by \( \pi/2 \) clockwise is \( \begin{bmatrix} 3 \ 5 \end{bmatrix} \).

Key Concepts

Vector Rotation

Vector rotation in 2D space can be visualized as turning a point around the origin on a coordinate plane. Imagine you have a vector like an arrow, starting from the origin (0,0) and pointing to (5, -3). By rotating the vector, you change its direction while keeping the arrow's base fixed at the origin.

To achieve this mathematically, we use a rotation matrix. A rotation matrix helps to systematically adjust the vector's components (its x and y parts). In our case, when we rotate a vector by an angle \( \theta \), we apply the rotation matrix:

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

This operation transforms the vector in the direction of the rotation without changing its length. This is particularly useful in graphics, robotics, and other fields where direction is crucial.

2D Transformations

2D transformations are operations that move or change objects within a two-dimensional plane. These transformations can include scaling, translation, and rotation, among others.

For rotation, specifically, the point or vector is rotated around a specified axis. In a 2D plane, this axis is the origin, (0,0). When performing a transformation, keeping track of angles is essential, as they determine how much rotation is applied.
Using the rotation matrix provides a straightforward method to achieve a precise transformation. By multiplying the original vector by the rotation matrix:

• We adjust the x and y coordinates of the vector
• Achieve a new orientation, while maintaining the vector's magnitude

Thus, 2D transformations, particularly rotations, can be highly useful for modeling, animations, and solving geometric problems.

Clockwise Rotation

When we talk about clockwise rotation, we refer to the direction in which the hands of a clock move, from the top to the right, then down, and finally back to the top again. For a mathematical rotation, the angle rotates in the same direction.
In the exercise, we rotate a vector by an angle of \( \frac{\pi}{2} \) clockwise, which is equivalent to -90 degrees. This specific operation shifts the position of our vector on the 2D plane. When we input an angle \( \theta = -\frac{\pi}{2} \) into the rotation matrix:

• \( \begin{bmatrix} \cos(-\frac{\pi}{2}) & \sin(-\frac{\pi}{2}) \ -\sin(-\frac{\pi}{2}) & \cos(-\frac{\pi}{2}) \end{bmatrix} = \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} \)

This matrix effectively rotates our vector to its new orientation, transforming it from \( \begin{bmatrix} 5 \ -3 \end{bmatrix} \) to \( \begin{bmatrix} 3 \ 5 \end{bmatrix} \). Such transformations are vital in understanding how objects can be re-positioned in digital environments or real-world tasks.