Problem 15 - University Calculus: Early Transcendentals

Question

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 \( \langle -\frac{\sqrt{3}}{2}, -\frac{1}{2} \rangle \).

Step 1: Understand the Rotation Matrix

When rotating a vector in a 2-dimensional plane, the new vector can be obtained using a rotation matrix. For a rotation by an angle \( \theta \) counterclockwise, the rotation matrix is given by \[R(\theta) = \begin{bmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{bmatrix}.\]We will use this matrix to find the rotated vector.

Step 2: Set Up the Rotation

We need to rotate the vector \( \langle 0, 1 \rangle \) by \( 120^{\circ} \). Convert this angle to radians since trigonometric functions typically use radians. \[ 120^{\circ} = \frac{2\pi}{3} \, \text{radians}. \]Using this angle, the rotation matrix becomes:\[R\left(\frac{2\pi}{3}\right) = \begin{bmatrix} \cos\left(\frac{2\pi}{3}\right) & -\sin\left(\frac{2\pi}{3}\right) \ \sin\left(\frac{2\pi}{3}\right) & \cos\left(\frac{2\pi}{3}\right) \end{bmatrix}.\]

Step 3: Compute Trigonometric Values

Calculate the cosine and sine of \( \frac{2\pi}{3} \). Recall:\[\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}, \\sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2}.\]

Step 4: Apply the Rotation Matrix

With these values, the rotation matrix is\[R\left(\frac{2\pi}{3}\right) = \begin{bmatrix} -\frac{1}{2} & -\frac{\sqrt{3}}{2} \ \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{bmatrix}.\]Apply this matrix to the vector \( \langle 0, 1 \rangle \):\[\begin{bmatrix} -\frac{1}{2} & -\frac{\sqrt{3}}{2} \ \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{bmatrix} \begin{bmatrix} 0 \ 1 \end{bmatrix} = \begin{bmatrix} -\frac{\sqrt{3}}{2} \ -\frac{1}{2} \end{bmatrix}.\]

Step 5: Verify Resulting Vector is a Unit Vector

A unit vector has a magnitude of 1. Calculate the magnitude of the resulting vector to verify:\[\sqrt{\left(-\frac{\sqrt{3}}{2}\right)^2 + \left(-\frac{1}{2}\right)^2} = \sqrt{\frac{3}{4} + \frac{1}{4}} = \sqrt{1} = 1.\]Thus, the resulting vector \( \langle -\frac{\sqrt{3}}{2}, -\frac{1}{2} \rangle \) is indeed a unit vector.

Key Concepts

Rotation Matrix

When dealing with rotating vectors, the rotation matrix is an essential tool. It helps us transform the direction and components of a vector by a certain angle in a given direction. In 2D plane rotations, if we want to rotate a vector by an angle \( \theta \) counterclockwise, we use the following matrix:

• \[ R(\theta) = \begin{bmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{bmatrix} \]

This matrix helps shift the vector while keeping its length the same, unless otherwise required. For example, when rotating \(120^\circ\) counterclockwise around the origin, substituting \(\theta\) with \( \frac{2\pi}{3} \) in radians (as angles are generally worked out in radians in such computations) yields the exact rotation matrix ready for calculation. The beauty and power of the rotation matrix lie in its consistent ability to rotate any given vector accurately.

Trigonometric Functions

Trigonometry provides the mathematics needed for understanding angles and their relationships in triangles. Rotation involves understanding sine and cosine functions since they relate to the angle's "projection" on the Cartesian coordinate axes.

• The cosine function \( \cos \theta \) describes the projection on the x-axis.
• The sine function \( \sin \theta \) details the y-axis projection.

For our specific example of a rotation by \( \frac{2\pi}{3} \):

• \( \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \)
• \( \sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2} \)

These trigonometric values are fundamental as they are embedded into the components of the rotation matrix. Understanding these basic trigonometric identities allows one to compute with confidence and accuracy.

Unit Vector

A unit vector plays a significant role in vector mathematics. It always has a magnitude or length of 1, serving as a building block for constructing other vectors of different lengths. When we rotate a vector such as \( \langle 0, 1 \rangle \) to obtain another vector, often the resulting vector is normalized (scaled) to become a unit vector.Verification of our rotated vector \( \langle -\frac{\sqrt{3}}{2}, -\frac{1}{2} \rangle \) being a unit vector involves verifying its magnitude through calculation:

• Magnitude: \( \sqrt{\left(-\frac{\sqrt{3}}{2}\right)^2 + \left(-\frac{1}{2}\right)^2} = \sqrt{1} = 1 \)

This calculation shows it maintains a length of 1, confirming its unit vector status.

Radians Conversion

Radians are commonly used in mathematics for angle measurements because they provide a natural way of describing angles. You convert degrees to radians using the simple conversion formula:

• \[ 120^{\circ} = \frac{120 \times \pi}{180} = \frac{2\pi}{3} \text{ radians} \]

Radians facilitate calculations in trigonometric functions and other mathematical operations as they link directly to the unit circle. The process of converting from degrees to radians is crucial when setting up problems involving rotational motion, ensuring calculations remain consistent with trigonometric functions that predominantly operate in radians.