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Chapter 12: Problem 13

Find the component form of the vector. The unit vector that makes an angle \(\theta=2 \pi / 3\) with the positive \(x\) -axis

Short Answer

Expert verified
The component form of the vector is \( \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right) \).

Step by step solution

01

Understanding the Unit Circle

To find the component form of the vector, we start by understanding what a unit vector is. A unit vector has a magnitude of 1. The angle given is \( \theta = \frac{2\pi}{3} \), which is in radians. It corresponds to 120 degrees on the unit circle.
02

Find the Components Using Trigonometry

On the unit circle, a vector making an angle \( \theta \) with the positive x-axis can be represented as \( (\cos(\theta), \sin(\theta)) \). Here, \( \theta = \frac{2\pi}{3} \). Therefore, the x-component of the vector is \( \cos\left(\frac{2\pi}{3}\right) \) and the y-component is \( \sin\left(\frac{2\pi}{3}\right) \).
03

Calculate the Components

Calculate the cosine component: \( \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \). Calculate the sine component: \( \sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2} \).
04

Write the Component Form

The vector in component form is \( \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right) \). This keeps the magnitude of the vector as 1, confirming it's a unit vector.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unit Circle
The unit circle is a fundamental concept in trigonometry and vector analysis. Imagine a circle with a radius of 1, centered at the origin of a coordinate system. That's the unit circle. It helps us visualize and calculate trigonometric functions: cosine and sine.
Let's break it down further:
  • For any angle \( \theta \) measured counterclockwise from the positive x-axis, the coordinates on the unit circle are \((\cos(\theta), \sin(\theta))\).
  • These coordinates correspond to the x and y components of a vector on the unit circle.
  • An angle \( \theta = \frac{2\pi}{3} \) radians corresponds to 120 degrees.
Understanding the unit circle helps in finding the component form of vectors, especially unit vectors, since they lie on this circle.
Trigonometry
When exploring vectors and unit circles, trigonometry plays a crucial role. It connects angles and ratios of triangle sides, which helps in determining positions on the unit circle.
In the context of vectors:
  • Cosine \( \cos(\theta) \) gives the x-coordinate of a point on the unit circle for an angle \( \theta \).
  • Sine \( \sin(\theta) \) gives the y-coordinate.
Let's put these to use:
  • For \( \theta = \frac{2\pi}{3} \):
    • \( \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \) is the x-component.
    • \( \sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2} \) is the y-component.
Trigonometry allows us to calculate these values to determine vector components.
Component Form
The component form of a vector expresses it as an ordered pair, indicating how far it travels in the x and y directions. For example, \((x, y)\) for a vector represents:
  • The x-component as how far it moves horizontally.
  • The y-component as how far it moves vertically.
For our unit vector with \( \theta = \frac{2\pi}{3} \):
  • The x-component \( = -\frac{1}{2} \)
  • The y-component \( = \frac{\sqrt{3}}{2} \)
By representing vectors this way, we can easily determine their direction and magnitude, making computations straightforward.
Unit Vector
In vector mathematics, a unit vector is crucial because it has a magnitude of 1 and provides direction. It lies on the unit circle due to its consistent magnitude.
Here's why unit vectors matter:
  • They serve as building blocks for defining directions in space.
  • In applications like physics, they simplify calculations by providing direction without changing a vector's strength.
The previous calculations confirm the vector \( \left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right) \) retains a magnitude of 1, embodying the properties of a unit vector.
Remember, whenever you're working with vectors on the unit circle, you're essentially dealing with unit vectors, making them pivotal in understanding directions and components.

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