Problem 14 - Thomas Calculus

Question

In Exercises \(9-16,\) find the component form of the vector. The unit vector that makes an angle \(\theta=-3 \pi / 4\) with the positive \(x\) -axis

Step-by-Step Solution

Answer

The component form is \( \langle -\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \rangle \).

Step 1: Understand the Unit Vector

A unit vector is a vector with a magnitude of 1. We are asked to find the component form of such a vector that makes a specified angle with the positive x-axis.

Step 2: Apply Trigonometric Functions

For any vector making an angle \( \theta \) with the positive x-axis, its component form can be found using cosine and sine. Here, the x-component is \( \cos(\theta) \) and the y-component is \( \sin(\theta) \).

Step 3: Calculate the X-Component

Substitute \( \theta = -3\pi/4 \) into the cosine function: \( \cos(-3\pi/4) = -\cos(\pi/4) = -\frac{\sqrt{2}}{2} \). This gives the x-component of the vector.

Step 4: Calculate the Y-Component

Substitute \( \theta = -3\pi/4 \) into the sine function: \( \sin(-3\pi/4) = -\sin(\pi/4) = -\frac{\sqrt{2}}{2} \). This gives the y-component of the vector.

Step 5: Formulate the Component Vector

Combine both components into a single vector. The vector's component form is \( \langle -\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \rangle \).

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. In this problem, we primarily use the sine and cosine functions to determine the components of a unit vector.
For a given angle \(\theta\), the cosine function \(\cos(\theta)\) gives the adjacent side's length of a right triangle relative to the hypotenuse. Similarly, the sine function \(\sin(\theta)\) provides the opposite side's length.
These trigonometric functions help in analyzing vectors because they can represent any vector in terms of its horizontal (x-axis) and vertical (y-axis) components. When constructing vectors, these functions enable us to translate an angle into directional movement along the x and y axes.

Vector Components

Vector components are the projections of a vector onto the x and y axes. By breaking a vector down into these components, we can express it more conveniently for various calculations.
In the discussion of unit vectors, which have a magnitude of 1, these components allow us to specify exactly how much of the vector points in the x-direction and how much in the y-direction.
Components are calculated using trigonometric functions: the x-component uses the cosine function \(\cos(\theta)\), while the y-component uses the sine function \(\sin(\theta)\). For instance, if a vector makes an angle of \(-3\pi/4\) with the x-axis:

• The x-component is \(\cos(-3\pi/4)\).
• The y-component is \(\sin(-3\pi/4)\).

This process forms the vector \(\langle \cos(\theta), \sin(\theta)\rangle\), combining both components.

Angle with x-axis

Angles with the x-axis allow us to understand a vector's direction relative to a standard reference line. The x-axis is often used as a baseline for measuring angles because it simplifies the comparison and computation of vectors.
In our scenario, the angle \(-3\pi/4\) is measured from the positive x-axis. Negative angles indicate clockwise rotation from the x-axis, while positive angles indicate counterclockwise rotation.
This specific angle pushes the vector into a different quadrant on the coordinate plane, affecting the signs of the vector components. Knowing the angle also guides the use of trigonometric identities like \(\cos(-\theta) = \cos(\theta)\) and \(\sin(-\theta) = -\sin(\theta)\) to find accurate component values.
Therefore, understanding an angle's interaction with the x-axis is vital for correctly determining vector directions and components.