Problem 13 - Thomas Calculus

Question

In Exercises \(9-16,\) find the component form of the vector. The unit vector that makes an angle \(\theta=2 \pi / 3\) with the positive \(x\) -axis

Step-by-Step Solution

Answer

The component form is \(-\frac{1}{2}\hat{i} + \frac{\sqrt{3}}{2}\hat{j}\).

Step 1: Understanding the Unit Vector

A unit vector is a vector with a magnitude of 1. We want to find the unit vector that makes an angle of \(\theta = \frac{2\pi}{3}\) radians with the positive x-axis.

Step 2: Calculating the Components

To find the components of the unit vector, we use the cosine and sine functions. The component form of the unit vector is given by \((\cos(\theta), \sin(\theta))\).

Step 3: Substitute the Angle

Substitute \(\theta = \frac{2\pi}{3}\) into the equations: \(\cos\left(\frac{2\pi}{3}\right)\) and \(\sin\left(\frac{2\pi}{3}\right)\).

Step 4: Evaluate the Trigonometric Functions

Calculate \(\cos\left(\frac{2\pi}{3}\right)\) which equals \(-\frac{1}{2}\), and \(\sin\left(\frac{2\pi}{3}\right)\) which equals \(\frac{\sqrt{3}}{2}\).

Step 5: Form the Component Vector

The component form of the unit vector is \(-\frac{1}{2}\hat{i} + \frac{\sqrt{3}}{2}\hat{j}\).

Key Concepts

Unit Vectors

A unit vector is a special type of vector, used extensively in mathematics and physics, especially in vector algebra. It is a vector that has a magnitude of exactly one unit. The fundamental characteristic of unit vectors is that they only indicate direction, not magnitude. This makes them so versatile in various calculations and transformations.

Some of the important points about unit vectors are:

• The standard unit vectors are typically denoted as \( \hat{i}, \hat{j}, \text{and} \hat{k} \) in 3-dimensional space.
• The direction of the unit vector is the same as the original vector but with a magnitude scaled down to 1.
• They are used as a basis to express any other vector. Meaning any vector can be represented as a combination of unit vectors.

In practical terms, if you have any vector \( \mathbf{v} = (a, b) \), its corresponding unit vector can be found by dividing each component by the magnitude of \( \mathbf{v} \). The formula for a unit vector \( \mathbf{u} \) in 2D is given as \( \left( \frac{a}{\|\mathbf{v}\|}, \frac{b}{\|\mathbf{v}\|} \right) \).

Through finding unit vectors, you redirect focus from the size of a vector to its direction, making them very useful for various applications such as in physics and computer graphics.

Trigonometric Functions

Trigonometric functions are key mathematical tools that relate angles to ratios of a right-angled triangle's sides. These functions are deeply rooted in vector algebra, especially when determining the direction and components of vectors.

For any unit vector making an angle \( \theta \) with the positive x-axis, the trigonometric functions \( \cos(\theta) \) and \( \sin(\theta) \) are used to express its components. This is due to the fact that:

• The cosine function \( \cos(\theta) \) gives the projection of the vector onto the x-axis.
• The sine function \( \sin(\theta) \) gives the projection of the vector onto the y-axis.

These functions are especially useful for rotating vectors and understanding their angular relationships. In our case, the angle \( \theta \) is \( \frac{2\pi}{3} \) radians, a common angle that represents two-thirds of a circle. By substituting this angle into the cosine and sine functions, Thus, trigonometric functions are indispensable for determining vector components when dealing with angles.

Component Form of Vectors

Vectors are often described in terms of components, simplifying calculations and making it easier to understand vector operations such as addition and subtraction. The component form of a vector breaks it down into individual parts, typically along the x and y axes in two-dimensional space.

The component form is essential for visualizing vectors:

• They are expressed as an ordered pair \( (x, y) \), where \( x \) and \( y \) are the magnitudes of the vector projected onto the respective axes.
• This format allows for easier manipulation and understanding of the vector's behavior.
• It simplifies addition and scalar multiplication of vectors. For example, adding two vectors becomes a simple addition of their corresponding components.

In the context of the unit vector with an angle \( \theta = \frac{2\pi}{3} \), its component form is derived using the trigonometric values. Thus, the vector is \[ -\frac{1}{2}\hat{i} + \frac{\sqrt{3}}{2}\hat{j} \].Here, the component form directly shows how the unit vector is directed when considering its angle with the x-axis. Hence, understanding the component form is crucial for practical applications in physics and engineering, where forces and velocities are often described in terms of their components.