Problem 46 - Trigonometry

Question

For each pair of vectors, find \(\mathbf{U}+\mathbf{V}, \mathbf{U}-\mathbf{V}\), and \(3 \mathbf{U}+2 \mathbf{V}\). $$\mathbf{U}=\mathbf{i}+4 \mathbf{j}, \mathbf{V}=7 \mathbf{i}-\mathbf{j}$$

Step-by-Step Solution

Answer

\( \mathbf{U} + \mathbf{V} = 8\mathbf{i} + 3\mathbf{j}, \mathbf{U} - \mathbf{V} = -6\mathbf{i} + 5\mathbf{j}, 3\mathbf{U} + 2\mathbf{V} = 17\mathbf{i} + 10\mathbf{j} \).

Step 1: Understanding the Problem

We are given two vectors, \( \mathbf{U} = \mathbf{i} + 4\mathbf{j} \) and \( \mathbf{V} = 7\mathbf{i} - \mathbf{j} \). Our task is to calculate \( \mathbf{U} + \mathbf{V} \), \( \mathbf{U} - \mathbf{V} \), and \( 3\mathbf{U} + 2\mathbf{V} \). Each operation will involve basic vector addition, subtraction, and scalar multiplication.

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

To add vectors \( \mathbf{U} \) and \( \mathbf{V} \), we add their respective components:\[ \mathbf{U} + \mathbf{V} = (\mathbf{i} + 4\mathbf{j}) + (7\mathbf{i} - \mathbf{j}) = (\mathbf{i} + 7\mathbf{i}) + (4\mathbf{j} - \mathbf{j}) = 8\mathbf{i} + 3\mathbf{j}. \]

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

To subtract \( \mathbf{V} \) from \( \mathbf{U} \), we subtract the corresponding components:\[ \mathbf{U} - \mathbf{V} = (\mathbf{i} + 4\mathbf{j}) - (7\mathbf{i} - \mathbf{j}) = (\mathbf{i} - 7\mathbf{i}) + (4\mathbf{j} + \mathbf{j}) = -6\mathbf{i} + 5\mathbf{j}. \]

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

First, perform the scalar multiplication of each vector, then sum the results:\[ 3\mathbf{U} = 3(\mathbf{i} + 4\mathbf{j}) = 3\mathbf{i} + 12\mathbf{j}, \]\[ 2\mathbf{V} = 2(7\mathbf{i} - \mathbf{j}) = 14\mathbf{i} - 2\mathbf{j}. \]Now add these two results:\[ 3\mathbf{U} + 2\mathbf{V} = (3\mathbf{i} + 12\mathbf{j}) + (14\mathbf{i} - 2\mathbf{j}) = 17\mathbf{i} + 10\mathbf{j}. \]

Key Concepts

Understanding Vector Addition

Vector addition is a fundamental concept in vector arithmetic. It involves adding two or more vectors to get a resultant vector. Vectors are quantities that have both magnitude and direction, often represented in terms of their components. To add vectors, you simply add their corresponding components. For example, if you have vector \( \mathbf{U} = a\mathbf{i} + b\mathbf{j} \) and vector \( \mathbf{V} = c\mathbf{i} + d\mathbf{j} \), the sum \( \mathbf{U} + \mathbf{V} \) is calculated as follows:

• Add the \( \mathbf{i} \) components: \( a + c \).
• Add the \( \mathbf{j} \) components: \( b + d \).

This results in a new vector \( \mathbf{U} + \mathbf{V} = (a+c)\mathbf{i} + (b+d)\mathbf{j} \). Each operation is performed separately for each component, making vector addition a straightforward process.In practical terms, this means breaking the vectors down into their individual components and combining them one-by-one. This method retains the direction and adds their corresponding magnitudes.

Grasping Vector Subtraction

Vector subtraction is not too different from vector addition but involves subtraction of the corresponding components. When you subtract vector \( \mathbf{V} \) from vector \( \mathbf{U} \), it can be visualized as adding \( -\mathbf{V} \) (the opposite direction of \( \mathbf{V} \)) to \( \mathbf{U} \).If \( \mathbf{U} = a\mathbf{i} + b\mathbf{j} \) and \( \mathbf{V} = c\mathbf{i} + d\mathbf{j} \), then the subtraction \( \mathbf{U} - \mathbf{V} \) yields:

• Subtract the \( \mathbf{i} \) components: \( a - c \).
• Subtract the \( \mathbf{j} \) components: \( b - d \).

Thus, \( \mathbf{U} - \mathbf{V} = (a-c)\mathbf{i} + (b-d)\mathbf{j} \).This subtraction process is like adding a vector that has been multiplied by \(-1\) to reverse its direction. It helps to think of vector subtraction as "adding the opposite." This concept is essential in fields like physics and engineering where determining the difference in vector quantities is necessary.

Essential Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a real number). This operation scales the vector, changing its magnitude while keeping its direction unchanged.Given a vector \( \mathbf{U} = a\mathbf{i} + b\mathbf{j} \) and a scalar \( k \), the result of scalar multiplication \( k\mathbf{U} \) is:

• Multiply each component of the vector by \( k \): \( k \cdot a\mathbf{i} \) and \( k \cdot b\mathbf{j} \).

This results in a new vector \( k\mathbf{U} = (ka)\mathbf{i} + (kb)\mathbf{j} \).The direction of the vector remains the same, but its length is stretched by the scalar factor. If \( k \) is negative, the vector also switches direction. Scalar multiplication is useful for adjusting the magnitude of vectors, allowing for flexibility in applications like physics where scaling force, velocity, or other vector quantities is common.