Problem 43 - Calculus for Biology and Medicine

Question

Use a rotation matrix to rotate the vector \(\left[\begin{array}{l}5 \\\ 2\end{array}\right]\) clockwise by the angle \(45^{\circ}\).

Step-by-Step Solution

Step 1: Understand the Rotation Matrix

For a 2D vector rotation clockwise by an angle \(\theta\), we use the rotation matrix: \[ R(\theta) = \begin{bmatrix} \cos(\theta) & \sin(\theta) \ -\sin(\theta) & \cos(\theta) \end{bmatrix} \] which, for a clockwise \(45^{\circ}\) rotation, becomes:\[ R(45^{\circ}) = \begin{bmatrix} \cos(45^{\circ}) & \sin(45^{\circ}) \ -\sin(45^{\circ}) & \cos(45^{\circ}) \end{bmatrix} \].

Key Concepts

Vector Rotation

Rotating a vector in 2D space is a useful mathematical operation, especially in applications like computer graphics and robotics. Imagine directing an arrow from the origin to a point in space, this is your vector. Rotation changes this arrow's direction without changing its length. To achieve this rotation by a specific angle, a **rotation matrix** is used. It provides a systematic method of changing the orientation of a vector. When you apply the rotation matrix to a vector, the vector's components are adjusted while keeping the overall magnitude constant. The beauty of vector rotation lies in its ability to maintain the figure's integrity while shifting its position. In essence, vector rotations are pivotal for transitioning graphics neatly without distortion.

Trigonometric Functions

Trigonometric functions describe relationships between the angles and sides of triangles. In the context of vector rotation, two main functions play crucial roles: - **Cosine** (\( \cos \theta \)): this function helps determine how much of the vector's movement aligns with the x-axis. It remains consistent whether the vector rotates forward or backward.- **Sine** (\( \sin \theta \)): this function defines the vector’s translation along the y-axis.For instance, during vector rotation in the clockwise direction by \( 45^{\circ} \), both cosine and sine will be evaluated at \( 45^{\circ} \), which equals \( \frac{\sqrt{2}}{2} \). This means that both components of the vector rotate equally from the x and y axes due to their equal magnitudes of cosine and sine at \( 45^{\circ} \).Understanding these functions provides a straightforward way to manipulate vector components in rotations, facilitating smooth transitions in a variety of 2D transformations.

2D Transformations

2D transformations refer to altering objects' position, shape, size, or orientation in two-dimensional space. They form the backbone of modifying graphics, images, or animations on a flat plane. The principal types of transformations include:

• **Translation**: Moving every point of an object a certain distance in a specified direction.
• **Scaling**: Increasing or decreasing the size of an object, either uniformly or non-uniformly.
• **Rotation**: Pivoting an object around a point, often the origin, through a specific angle.

In the given exercise, we focus on rotation, a transformation that modifies the angular position of an object but retains its shape and dimensions. The rotation matrix applies this principle by updating a vector's orientation. By following a clocklike motion, a clockwise rotation translates all points accordingly, adhering to the implemented angle of rotation. This transformation is integral in various fields such as physics simulations, graphical animations, and more, ensuring that objects move precisely while maintaining their fundamental properties.