Problem 57 - Algebra and Trigonometry

Question

In Exercises \(57-60,\) let $$\begin{aligned}\mathbf{u} &=a_{1} \mathbf{i}+b_{1} \mathbf{j} \\\\\mathbf{v} &=a_{2} \mathbf{i}+b_{2} \mathbf{j} \\\\\mathbf{w} &=a_{3} \mathbf{i}+b_{3} \mathbf{j}\end{aligned} $$ Prove each property by obtaining the vector on each side of the equation. Have you proved a distributive, associative, or commutative property of vectors? $$ \mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u} $$

Step-by-Step Solution

Answer

As observed, the order in which vectors \( \mathbf{u} \) and \( \mathbf{v} \) are added doesn't change the result. This observation confirms the validity of the commutative property of vectors: \( \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} \)

Step 1: Describe vectors in component form

Decompose vectors \( \mathbf{u} \) and \( \mathbf{v} \) into their respective components. \( \mathbf{u} \) is represented as \( a_{1}\mathbf{i} + b_{1}\mathbf{j} \) and \( \mathbf{v} \) as \( a_{2}\mathbf{i} + b_{2}\mathbf{j} \)

Step 2: Calculate \(\mathbf{u} + \mathbf{v}\)

Add vectors \( \mathbf{u} \) and \( \mathbf{v} \) by adding their respective components: \( (a_{1} + a_{2})\mathbf{i} + (b_{1} + b_{2})\mathbf{j} \).

Step 3: Calculate \(\mathbf{v} + \mathbf{u}\)

Add vectors \( \mathbf{v} \) and \( \mathbf{u} \) by adding their respective components: \( (a_{2} + a_{1})\mathbf{i} + (b_{2} + b_{1})\mathbf{j} \).

Step 4: Compare the results

Compare the results of Steps 2 and 3. Upon comparison, it can be observed that \( (a_{1} + a_{2})\mathbf{i} + (b_{1} + b_{2})\mathbf{j} = (a_{2} + a_{1})\mathbf{i} + (b_{2} + b_{1})\mathbf{j} \). This proves the commutative property of vectors in the context of this particular exercise.

Key Concepts

Commutative Property

When working with vectors, it's essential to understand the commutative property, which is a fundamental principle in mathematics. This property tells us that the order in which we add numbers or vectors does not change the result. In the world of vectors, the commutative property of addition can be expressed as

• \( \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} \)

This means if you take two vectors, \( \mathbf{u} \) and \( \mathbf{v} \), and add them, the sum is the same no matter which one you add first. For our example, both \( \mathbf{u} + \mathbf{v} \) and \( \mathbf{v} + \mathbf{u} \) result in the vector

• \((a_{1} + a_{2})\mathbf{i} + (b_{1} + b_{2})\mathbf{j}\)

The components add up to the same number, confirming the commutative property. This holds true because between pairs of real numbers, addition is commutative, and vectors are built from these real numbers. No matter how we arrange the terms, the end result remains consistent.

Vector Addition

In mathematics, particularly within vector algebra, vector addition refers to the process of adding two or more vectors together to produce a resultant vector. The key idea of vector addition is understanding that similar components (like the \(i\)-components and \(j\)-components) are added together.The step-by-step method for adding vectors \( \mathbf{u} \) and \( \mathbf{v} \), given by the expressions:

• \( \mathbf{u} = a_{1}\mathbf{i} + b_{1}\mathbf{j}\)
• \( \mathbf{v} = a_{2}\mathbf{i} + b_{2}\mathbf{j}\)

When you perform vector addition, you add the \(i\)-components:

• \(a_{1} + a_{2}\)

And the \(j\)-components:

• \(b_{1} + b_{2}\)

The result is a new vector:

• \((a_{1} + a_{2})\mathbf{i} + (b_{1} + b_{2})\mathbf{j}\)

This approach ensures that we combine vectors in a way that respects both their magnitude and direction. Vector addition is not only applicable in 2D but can also be extended into 3D and beyond, where each vector involves additional components, like \(k\) for three dimensions.

Component Form of Vectors

The component form of a vector is a way to break down a vector into its fundamental parts. In two-dimensional space, any vector can be expressed as a combination of the unit vectors \( \mathbf{i} \) (along the x-axis) and \( \mathbf{j} \) (along the y-axis).Consider a vector \( \mathbf{u} \) given by:

• \( \mathbf{u} = a_{1}\mathbf{i} + b_{1}\mathbf{j} \)

Here, the numbers \( a_1 \) and \( b_1 \) are the components of \( \mathbf{u} \), representing its influence in the horizontal and vertical directions, respectively. The component form

• Facilitates calculations, such as addition, because each component can be treated separately.
• Makes it simpler to visualize the vector on a coordinate grid.

When vectors are expressed in component form, it allows for trivial addition, subtraction, and even scaling since you can carry out these operations on individual components. This makes handling vectors in physics, engineering, and computer science much more convenient. Understanding how to decompose vectors in this form allows for greater flexibility in mathematical manipulations and problem-solving.