Problem 15 - Thomas Calculus

Question

In Exercises \(9-16,\) find the component form of the vector. The unit vector obtained by rotating the vector \(\langle 0,1\rangle 120^{\circ}\) counterclockwise about the origin

Step-by-Step Solution

Answer

The unit vector is \( \langle -\frac{1}{2}, \frac{\sqrt{3}}{2} \rangle \).

Step 1: Understand the Problem

We need to find the component form of a unit vector obtained by rotating the vector \( \langle 0, 1 \rangle \) through an angle of \( 120^{\circ} \) counterclockwise about the origin.

Step 2: Identify Original Vector Components

The original vector is \( \langle 0, 1 \rangle \). This means it has an \( x \)-component of \( 0 \) and a \( y \)-component of \( 1 \).

Step 3: Setup Rotation Matrix

To rotate a vector counterclockwise by an angle \( \theta \), use the rotation matrix:\[R(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{bmatrix}\]For \( \theta = 120^{\circ} \), we have \( \cos(120^{\circ}) = -\frac{1}{2} \) and \( \sin(120^{\circ}) = \frac{\sqrt{3}}{2} \).

Step 4: Apply Rotation Matrix

Using the rotation matrix and the original vector, compute the new vector:\[\begin{bmatrix} \cos(120^{\circ}) & -\sin(120^{\circ}) \ \sin(120^{\circ}) & \cos(120^{\circ}) \end{bmatrix} \begin{bmatrix} 0 \ 1 \end{bmatrix} = \begin{bmatrix} -\frac{1}{2} \ \frac{\sqrt{3}}{2} \end{bmatrix}\]So the rotated vector is \( \langle -\frac{1}{2}, \frac{\sqrt{3}}{2} \rangle \).

Step 5: Ensure It's a Unit Vector

To check if the vector is a unit vector, we compute its magnitude:\[\sqrt{\left(-\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1\]Since the magnitude is 1, it is already a unit vector.

Key Concepts

Unit Vector

A unit vector is a vector that has a magnitude of 1. Unit vectors are essential in mathematics and physics for denoting directions without concerning magnitude. They serve as building blocks in vector calculations, allowing you to stretch or compress vectors along specific directions.

• To find a unit vector in the direction of a non-zero vector, you divide each component by the vector's magnitude.
• The magnitude of a unit vector is always 1, making it very convenient to work with.

In the context of the exercise, we wanted to ensure the rotated vector remains a unit vector. After applying the rotation, we verified the new vector had a magnitude of 1, confirming it is a unit vector in the new direction.

Rotation Matrix

The rotation of vectors in a plane is elegantly handled through the rotation matrix. This matrix allows you to rotate a vector by any angle \( \theta \) around the origin. Rotation matrices are fixed, providing consistent results no matter the input vector.

• The general form of the rotation matrix in two dimensions is: \\[ R(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{bmatrix} \]
• For our exercise, we used a rotation of \(120^{\circ}\), calculated using trigonometric values for that angle.

This matrix transforms an input vector \( \langle 0, 1 \rangle \) to a new one by applying simple linear algebra - matrix multiplication. Understanding how this matrix operates is crucial for mastering vector rotations.

Trigonometric Functions

Trigonometric functions are pivotal in creating and manipulating rotation matrices. They reveal profound relationships between angles and lengths, central to computations involving circles and oscillations.

• In rotation, the cosine function holds the role of preserving the original x-component while fitting the vector into the new directional component.
• The sine function complements cosine, adjusting the y-components accordingly, pivotal when shifting vectors in a plane.

For the angle 120 degrees used in our problem, \( \cos(120^{\circ}) = -\frac{1}{2} \) and \( \sin(120^{\circ}) = \frac{\sqrt{3}}{2} \). These specific values are derived from the symmetrical properties of angles and triangles within the unit circle, offering precise control over vector manipulation when embedded in the rotation matrix.