Problem 44 - Calculus for Biology and Medicine

Question

Use a rotation matrix to rotate the vector \(\left[\begin{array}{l}-2 \\\ -3\end{array}\right]\) counterclockwise by the angle \(45^{\circ}\).

Step-by-Step Solution

Answer

The rotated vector is \( \begin{bmatrix} \frac{\sqrt{2}}{2} \\ \frac{-5\sqrt{2}}{2} \end{bmatrix} \).

Step 1: Define the Rotation Matrix

The rotation matrix for counterclockwise rotation by an angle \( \theta \) is given by:\[R(\theta) = \begin{bmatrix}\cos \theta & -\sin \theta \\sin \theta & \cos \theta\end{bmatrix}\]For \( \theta = 45^{\circ} \), we calculate \( \cos 45^{\circ} = \sin 45^{\circ} = \frac{\sqrt{2}}{2} \). Thus, the rotation matrix becomes:\[R(45^{\circ}) = \begin{bmatrix}\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}\end{bmatrix}\]

Step 2: Multiply the Rotation Matrix with the Vector

Now, we apply the rotation matrix to the vector \( \begin{bmatrix} -2 \ -3 \end{bmatrix} \). The rotated vector \( \mathbf{v'} \) is computed as:\[\mathbf{v'} = R(45^{\circ}) \begin{bmatrix} -2 \ -3 \end{bmatrix}\]

Step 3: Calculate the Components of the New Vector

Perform the matrix multiplication:\[\begin{bmatrix}\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}\end{bmatrix}\begin{bmatrix} -2 \ -3 \end{bmatrix} = \begin{bmatrix}\frac{\sqrt{2}}{2} \times -2 + (-\frac{\sqrt{2}}{2}) \times -3 \\frac{\sqrt{2}}{2} \times -2 + \frac{\sqrt{2}}{2} \times -3\end{bmatrix}\]

Step 4: Simplify the Expressions

Simplify each component:1. \( \frac{\sqrt{2}}{2} \times -2 + (-\frac{\sqrt{2}}{2}) \times -3 = \frac{-2\sqrt{2} + 3\sqrt{2}}{2} = \frac{\sqrt{2}}{2} \)2. \( \frac{\sqrt{2}}{2} \times -2 + \frac{\sqrt{2}}{2} \times -3 = \frac{-2\sqrt{2} - 3\sqrt{2}}{2} = \frac{-5\sqrt{2}}{2} \)The new vector is:\[\begin{bmatrix} \frac{\sqrt{2}}{2} \ \frac{-5\sqrt{2}}{2} \end{bmatrix}\]

Step 5: Provide the Final Answer

The vector \( \begin{bmatrix} -2 \ -3 \end{bmatrix} \) rotated counterclockwise by \( 45^{\circ} \) is:\[\begin{bmatrix} \frac{\sqrt{2}}{2} \ \frac{-5\sqrt{2}}{2} \end{bmatrix}\]

Key Concepts

Vector Rotation

In mathematics, vector rotation involves turning a vector around the origin of a coordinate system while retaining its magnitude. This process is essential in fields like physics, engineering, and computer graphics. If you imagine a vector as an arrow pointing in a particular direction, rotating it means pivoting it around the origin to point in another direction.

When rotating vectors in two dimensions, the direction in which the vector is turned depends on the rotational angle. If we rotate a vector counterclockwise by a certain angle, the head of the vector moves to a new position on the plane, though its origin stays fixed. In our example, we rotate the vector \([-2, -3]\) by an angle of \(45^{\circ}\).

Understanding vector rotation is crucial, especially when working on problems involving rotational dynamics or computer simulations that require changing object orientations.

Matrix Multiplication

Matrix multiplication is a fundamental concept in linear algebra and is heavily utilized in operations involving transformations, including vector rotation. Specifically, when rotating vectors, the vector is multiplied by a rotation matrix, which alters its orientation in space.

To perform matrix multiplication, each element of the resulting matrix is obtained by calculating the dot product of corresponding rows and columns from the involved matrices. In the context of our vector \([-2, -3]\) and its rotation matrix:

• Multiply the first row of the rotation matrix by the column of the vector to get the first element of the new vector.
• Multiply the second row of the rotation matrix by the column of the vector to get the second element of the new vector.

The completed calculation in our example leads to a new vector \( \left[ \frac{\sqrt{2}}{2}, \frac{-5\sqrt{2}}{2} \right] \). Understanding how and why we multiply matrices is key to executing a proper rotational transformation.

Angle of Rotation

The angle of rotation is the measure in degrees or radians that determines how far and in which direction a vector is rotated. This angle is integral in defining the rotation matrix used for transforming the vector.

For our problem, the angle provided was \(45^{\circ}\). This means every point on the vector is shifted \(45^{\circ}\) counterclockwise around the origin. The choice of angle affects components like \(\cos(\theta)\) and \(\sin(\theta)\) in the rotation matrix, consequently altering the rotated vector's coordinates.

It’s helpful to remember that common angles, such as \(30^{\circ}\), \(45^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\), are often used because their trigonometric functions have simple, exact values. For a \(45^{\circ}\) angle, \(\cos(45^{\circ}) = \sin(45^{\circ}) = \frac{\sqrt{2}}{2}\), simplifying the process of creating and applying the rotation matrix. Recognizing these functions allows for smoother calculations and interpretations of geometric transformations.