Problem 47 - Trigonometry

Question

For each pair of vectors, find \(\mathbf{U}+\mathbf{V}, \mathbf{U}-\mathbf{V}\), and \(3 \mathbf{U}+2 \mathbf{V}\). $$\mathbf{U}=6 \mathbf{i}, \mathbf{V}=-8 \mathbf{j}$$

Step-by-Step Solution

Answer

\( \mathbf{U} + \mathbf{V} = 6 \mathbf{i} - 8 \mathbf{j}, \mathbf{U} - \mathbf{V} = 6 \mathbf{i} + 8 \mathbf{j}, 3\mathbf{U} + 2\mathbf{V} = 18\mathbf{i} - 16\mathbf{j} \).

Step 1: Understanding the Vectors

The vector \( \mathbf{U} \) is given as \( 6 \mathbf{i} \), which means it has a component of 6 along the x-axis and none along the y-axis (0 \( \mathbf{j} \)). The vector \( \mathbf{V} \) is given as \( -8 \mathbf{j} \), meaning it has no x-component and a component of -8 along the y-axis.

Step 2: Calculating \( \mathbf{U} + \mathbf{V} \)

To find \( \mathbf{U} + \mathbf{V} \), we add the corresponding components of \( \mathbf{U} \) and \( \mathbf{V} \). For the x-component, it's \( 6 \+ 0 = 6 \). For the y-component, it's \( 0 + (-8) = -8 \). Thus, \( \mathbf{U} + \mathbf{V} = 6 \mathbf{i} - 8 \mathbf{j} \).

Step 3: Calculating \( \mathbf{U} - \mathbf{V} \)

To find \( \mathbf{U} - \mathbf{V} \), subtract the components of \( \mathbf{V} \) from those of \( \mathbf{U} \). For the x-component, it's \( 6 - 0 = 6 \). For the y-component, it's \( 0 - (-8) = 8 \). Therefore, \( \mathbf{U} - \mathbf{V} = 6 \mathbf{i} + 8 \mathbf{j} \).

Step 4: Calculating \( 3\mathbf{U} + 2\mathbf{V} \)

First, calculate \( 3\mathbf{U} \) and \( 2\mathbf{V} \). \( 3\mathbf{U} = 3 \times 6 \mathbf{i} = 18\mathbf{i} \) and \( 2\mathbf{V} = 2 \times (-8) \mathbf{j} = -16\mathbf{j} \). Then add these results: \( 3\mathbf{U} + 2\mathbf{V} = 18\mathbf{i} - 16\mathbf{j} \).

Key Concepts

Vector Subtraction

Vector subtraction is the process of finding the difference between two vectors. This involves subtracting the corresponding components of one vector from another. Consider vectors: \\[ \mathbf{U} = 6\mathbf{i} \text{ and } \mathbf{V} = -8\mathbf{j} \].
To compute \(\mathbf{U} - \mathbf{V}\), we first identify the components:

• \( \mathbf{U} \) has components \( (6, 0) \) because it is entirely along the x-axis.
• \( \mathbf{V} \) has components \( (0, -8) \) because it is entirely along the y-axis.

Now, subtract \(\mathbf{V}\) from \(\mathbf{U}\):

• Subtract the x-components: \( 6 - 0 = 6 \)
• Subtract the y-components: \( 0 - (-8) = 8 \)

These calculations yield the result \(\mathbf{U} - \mathbf{V} = 6\mathbf{i} + 8\mathbf{j}\).
The subtraction effectively turns \(\mathbf{V}\) into a vector moving in the opposite direction during this process, which is why we add \(8\mathbf{j}\) when subtracting \( -8\mathbf{j} \).

Scalar Multiplication of Vectors

Scalar multiplication involves multiplying a vector by a scalar (a numerical value), which scales or resizes the vector without changing its direction. Let's consider the vectors \( \mathbf{U} \) and \( \mathbf{V} \) once more.
For \( 3\mathbf{U} \):

• Multiply the magnitude of \( \mathbf{U} \), which is 6, by the scalar 3: \( 3 \times 6 = 18 \)
• This results in \( 18 \mathbf{i} \), as \( \mathbf{U} \) lies along the x-axis.

For \( 2\mathbf{V} \):

• Multiply the magnitude of \( \mathbf{V} \), which is -8, by the scalar 2: \( 2 \times (-8) = -16 \)
• This results in \( -16 \mathbf{j} \), as \( \mathbf{V} \) lies along the y-axis.

The vectors are scaled according to the scalar, increasing or decreasing their length proportionally. However, the vectors do not change direction unless the scalar is negative, which inverts their direction.
Thus, when calculating \( 3\mathbf{U} + 2\mathbf{V} \), we add the scaled vectors like before:

• Combine \( 18 \mathbf{i} \) and \( -16 \mathbf{j} \) to form the result \( 3\mathbf{U} + 2\mathbf{V} = 18 \mathbf{i} - 16 \mathbf{j} \).

Vector Components

Vectors are quantities that have both magnitude and direction and are often represented in two parts: the x-component and the y-component, when considering 2D vectors. These components help break down the vectors into simpler parts for calculations.
In the exercise, the vector \( \mathbf{U} = 6 \mathbf{i} \) can be viewed like this:

• x-component: 6
• y-component: 0

This means \( \mathbf{U} \) has a strength of 6 in the x-direction, with no movement in the y-direction.
Similarly, \( \mathbf{V} = -8 \mathbf{j} \):

• x-component: 0
• y-component: -8

This indicates \( \mathbf{V} \) has no strength in the x-direction but moves 8 units in the negative y-direction.
Understanding vector components is crucial because it allows us to perform operations such as vector addition and subtraction. By analyzing the components, it becomes easier to manipulate vectors mathematically, helping in various applications like physics and engineering.
Components act as the building blocks of any vector operation, providing a straightforward means to achieve desired results.