Problem 13 - Precalculus

Question

Assume that the vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c},\) and \(\mathbf{d}\) are defined as follows: $$\mathbf{a}=\langle 2,3\rangle \quad \mathbf{b}=\langle 5,4\rangle \quad \mathbf{c}=\langle 6,-1\rangle \quad \mathbf{d}=\langle-2,0\rangle$$ Compute each of the indicated quantities. $$\mathbf{a}+\mathbf{b}$$

Step-by-Step Solution

Answer

\(\mathbf{a} + \mathbf{b} = \langle 7, 7 \rangle\)

Step 1: Identify the Components of Each Vector

We start by identifying the components of vectors \(\mathbf{a}\) and \(\mathbf{b}\). The vector \(\mathbf{a}\) has components \((2, 3)\), and vector \(\mathbf{b}\) has components \((5, 4)\).

Step 2: Add the Corresponding Components

Add the corresponding components of vectors \(\mathbf{a}\) and \(\mathbf{b}\). The sum of the x-components is \(2 + 5\), and the sum of the y-components is \(3 + 4\).

Step 3: Perform the Calculation

Calculate the sums: \(2 + 5 = 7\) and \(3 + 4 = 7\).

Step 4: Write the Resultant Vector

Combine the sums from the previous step to write the resultant vector: \(\mathbf{a} + \mathbf{b} = \langle 7, 7 \rangle\).

Key Concepts

Component-wise Addition

When adding vectors, the process is much like working with numbers, but with an extra twist because vectors have multiple components. To make things simpler, we use component-wise addition.
First, it's crucial to understand that each vector is made up of components. For instance, vector \( \mathbf{a} \) is noted as \( \langle 2, 3 \rangle \). This means it has two components: 2 in the x-direction and 3 in the y-direction. Similarly, vector \( \mathbf{b} \) has components \( \langle 5, 4 \rangle \).
The key idea in component-wise addition is:

• Add the x-components together.
• Add the y-components together.

In our exercise, we add the x-components of \( \mathbf{a} \) and \( \mathbf{b} \), which is \( 2 + 5 \). Next, we add the y-components: \( 3 + 4 \). This method allows you to easily find the components of the resultant vector, simplifying vector addition without getting mixed up in the details of the direction each vector points.

Resultant Vector

Once you've added the components, you arrive at what is known as the resultant vector. Think of it as the 'final sum' or the 'total' when adding two or more vectors together.
In the exercise, after performing component-wise addition, we calculated that the resultant x-component is 7 and the resultant y-component is also 7. This gives us the resultant vector \( \mathbf{a} + \mathbf{b} = \langle 7, 7 \rangle \).
The resultant vector represents the total effect of adding the individual vectors together. It points in a direction that is a balance of all the initial vectors involved and can be visualized as a single vector that achieves the same outcome as combining the individual vectors.

Vectors in Two Dimensions

Vectors help us represent quantities that have both a magnitude and a direction, making them particularly useful in two dimensions (2D). With 2D vectors, you can accurately describe movement or forces in a plane, such as the motion of a car moving east and north simultaneously.
Each vector in 2D is defined by two components: an x-component (horizontal direction) and a y-component (vertical direction). The vector represents movement along these two axes.
When visualizing 2D vectors:

• They can be drawn as arrows on a coordinate grid.
• The length of the arrow represents the magnitude of the vector.
• The angle shows the direction.

In our example, vector \( \mathbf{a} \) with components \( \langle 2, 3 \rangle \) can be thought of as moving 2 units along the x-axis and 3 units upwards along the y-axis. This helps provide a clear picture of how vectors behave on graphs and allows physicists and engineers to make precise calculations. Understanding vectors in two dimensions is fundamental to mastering more complex topics in physics and engineering.