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Chapter 7: Problem 64

Perform the indicated vector operation. $$ (4 \mathbf{i}-2 \mathbf{j})+(3 \mathbf{i}-5 \mathbf{j}) $$

Short Answer

Expert verified
The resultant vector is \(7 \mathbf{i} - 7 \mathbf{j}\).

Step by step solution

01

Understanding the Problem

Our task is to perform the vector addition of two vectors given in component form: \(4 \mathbf{i} - 2 \mathbf{j}\) and \(3 \mathbf{i} - 5 \mathbf{j}\). This involves adding corresponding components (\(\mathbf{i}\) and \(\mathbf{j}\) components separately).
02

Adding the \(\mathbf{i}\)-Components

Identify the \(\mathbf{i}\)-components of both vectors. The first vector has an \(\mathbf{i}\)-component of 4, and the second vector has an \(\mathbf{i}\)-component of 3. Add them together: \[ 4 + 3 = 7 \]
03

Adding the \(\mathbf{j}\)-Components

Identify the \(\mathbf{j}\)-components of both vectors. The first vector has a \(-2\mathbf{j}\) component, and the second vector has a \(-5\mathbf{j}\) component. Add them together: \[ -2 + (-5) = -7 \]
04

Writing the Resultant Vector

Combine the results of the previous steps to form the resultant vector. The resultant vector is formed by combining the \(\mathbf{i}\)-component and the \(\mathbf{j}\)-component: \[ 7 \mathbf{i} - 7 \mathbf{j} \]

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Key Concepts

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

Component Form
Vector operations, like addition, are easier when vectors are expressed in their **component form**. This means we separate the vector into its horizontal and vertical parts. Instead of a single vector with magnitude and direction, we break it down into how far it goes along the x-axis (horizontal) and along the y-axis (vertical).
  • The horizontal part is represented by the
  • The vertical part is represented by the j-component.
For example, the vector \(4 \mathbf{i} - 2 \mathbf{j}\) represents moving 4 units on the horizontal and -2 units on the vertical. Writing vectors this way helps you visually understand and perform vector operations by simply dealing with these separated components.
i-component
The **i-component** of a vector represents the horizontal part of the vector. It tells us how much the vector moves along the x-axis. When you see a vector such as \(4 \mathbf{i}\), it means that the vector moves 4 units to the right.
In the expression \( (4 \mathbf{i} - 2 \mathbf{j}) + (3 \mathbf{i} - 5 \mathbf{j}) \),
  • The first vector contributes a +4 to the i-component.
  • The second vector contributes a +3 to the i-component.
To find the total i-component, simply add these values together: \[4 + 3 = 7\]This means the resultant vector has an i-component of \(7\), indicating a horizontal movement of 7 units to the right.
j-component
The **j-component** addresses the vertical part of a vector. It represents how much the vector moves along the y-axis. For instance, in \(-2 \mathbf{j}\), the vector moves downwards by 2 units.
Considering our operation \( (4 \mathbf{i} - 2 \mathbf{j}) + (3 \mathbf{i} - 5 \mathbf{j}) \),
  • The first vector has a j-component of -2.
  • The second vector has a j-component of -5.
Adding these values gives you the total j-component:\[-2 + (-5) = -7 \]This shows that the resultant vector goes 7 units downward because of the negative sign in the j-component.
Resultant Vector
A **resultant vector** is what you get after performing a vector operation, such as addition. It shows the final direction and magnitude resulting from combining the effects of all vectors involved.
In our exercise, we have added the i-components and j-components separately:
  • i-component: \(4 + 3 = 7\)
  • j-component: \(-2 + (-5) = -7\)
By putting together these results, we form the new vector:\[7 \mathbf{i} - 7 \mathbf{j}\]The resultant vector \(7 \mathbf{i} - 7 \mathbf{j}\) demonstrates that after adding the vectors, the total movement is 7 units right (horizontal) and 7 units down (vertical). Understanding this is crucial in visualizing how vectors combine in real-world scenarios.

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Most popular questions from this chapter

Find the angle \(\theta\left(0^{\circ} \leq \theta \leq 180\right.\); round to the nearest degree) between each pair of vectors. $$ \langle-6,2\rangle \text { and }\langle-3,8\rangle $$

Find each of the following dot products. $$ \langle 0.8,-0.5\rangle \cdot\langle 2,6\rangle $$

Find each of the following dot products. $$ \langle 4 \sqrt{2}, \sqrt{7}\rangle \cdot\langle-\sqrt{2},-\sqrt{7}\rangle $$

Find each of the following dot products. $$ \langle 5,-2\rangle \cdot\langle-1,-1\rangle $$

Find the angle \(\theta\left(0^{\circ} \leq \theta \leq 180\right.\); round to the nearest degree) between each pair of vectors. $$ \langle-4,6\rangle \text { and }\langle-6,8\rangle $$
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