Problem 27 - A Graphical Approach to Precalculus with Limits

Question

Given \(\mathbf{u}=\langle- 2,5\rangle\) and \(\mathbf{v}=\langle 4,3\rangle,\) find each vector. Do not use a calculator. $$\mathbf{u}+\mathbf{v}$$

Step-by-Step Solution

Answer

\( \mathbf{u} + \mathbf{v} = \langle 2, 8 \rangle \).

Step 1: Understanding the Problem

You are given two vectors: \( \mathbf{u} = \langle -2, 5 \rangle \) and \( \mathbf{v} = \langle 4, 3 \rangle \). You need to find the vector resulting from their addition, \( \mathbf{u} + \mathbf{v} \).

Step 2: Vector Addition Formula

To add two vectors \( \mathbf{u} \) and \( \mathbf{v} \), use the formula \( \mathbf{u} + \mathbf{v} = \langle u_1 + v_1, u_2 + v_2 \rangle \), where \( u_1 \) and \( u_2 \) are the components of \( \mathbf{u} \) and \( v_1 \) and \( v_2 \) are the components of \( \mathbf{v} \).

Step 3: Component-wise Addition

Identify the components from \( \mathbf{u} \) and \( \mathbf{v} \): \( u_1 = -2, u_2 = 5, v_1 = 4, v_2 = 3 \). Add these components together: \( u_1 + v_1 = -2 + 4 = 2 \) and \( u_2 + v_2 = 5 + 3 = 8 \).

Step 4: Constructing the Resultant Vector

Using the added components, construct the resultant vector: \( \mathbf{u} + \mathbf{v} = \langle 2, 8 \rangle \).

Key Concepts

Component-wise Addition

When we talk about adding vectors component-wise, we essentially break down each vector into its parts, known as components, and add them individually. Each vector in two-dimensional space is expressed with two components. For instance, the vector \( \mathbf{u} = \langle -2, 5 \rangle \) has components \(-2\) and \(5\), and \( \mathbf{v} = \langle 4, 3 \rangle \) has components \(4\) and \(3\).

• Identify the first components: \( u_1 = -2 \) and \( v_1 = 4 \).
• Add these together to get \( u_1 + v_1 = -2 + 4 = 2 \).
• Identify the second components: \( u_2 = 5 \) and \( v_2 = 3 \).
• Add these together to get \( u_2 + v_2 = 5 + 3 = 8 \).

Breaking down vectors like this makes the math far easier to handle, especially in higher dimensions.

Resultant Vector

Once you've completed the component-wise addition, you can construct the new vector called the resultant vector. This vector is essentially the sum of the original vectors, \( \mathbf{u} \) and \( \mathbf{v} \).

The process goes like this:

• You've found the newly calculated components: \(2\) for the first component and \(8\) for the second component.
• Place these components into the vector form \(\langle x, y \rangle\), giving you \(\langle 2, 8 \rangle\).

The resultant vector \( \mathbf{u} + \mathbf{v} = \langle 2, 8 \rangle \) captures the net effect of combining the original vectors. It's important to remember that this process can be extended to any dimension, simply by continuing to add corresponding components.

Vectors

Vectors are mathematical objects that hold both a magnitude and a direction. They are incredibly useful in a variety of fields such as physics, engineering, and computer science. In two-dimensional space, vectors are often represented by two numbers, illustrating their components.

• The vectors \( \mathbf{u} = \langle -2, 5 \rangle \) and \( \mathbf{v} = \langle 4, 3 \rangle \) are examples of simple 2D vectors.
• Each component of a vector specifies a direction in a particular dimension; the first is along the x-axis and the second is along the y-axis.
• Addition of vectors as seen here combines their effects, resulting in a new vector that takes into account both original magnitudes and directions.

Embracing the concept of vectors allows you to solve complex problems involving multiple directions in a very straightforward manner.