91Ó°ÊÓ

Prove that any two of the following expressions are equal: $$ \begin{aligned} &\mathbf{v}+(\mathbf{w}+(\mathbf{x}+\mathbf{y})) \\ &\mathbf{v}+((\mathbf{w}+\mathbf{x})+\mathbf{y}) \\ The point is that even though addition of vectors always combines exactly two vectors to produce a third, it doesn't really matter which pairs are added together first. Consequently, it is customary to eliminate the parentheses when dealing with sums of vectors. The vector represented by any one of the five expressions above will be denoted \(\mathbf{v}+\mathbf{w}+\mathbf{x}+\mathbf{y}\). To show why this sloppiness is legitimate when an arbitrary number of vectors is to be summed would require an elaborate system for keeping track of parentheses. If you are familiar with proofs by induction, you may want to regard this as a challenge. In any case, we want to be free from writing all these parentheses. From now on we will rely on our happy experiences with real numbers where a similar problem is ignored. &(\mathbf{v}+\mathbf{w})+(\mathbf{x}+\mathbf{y}) \\ &(\mathbf{v}+(\mathbf{w}+\mathbf{x}))+\mathbf{y} \\ &((\mathbf{v}+\mathbf{w})+\mathbf{x})+\mathbf{v} \end{aligned} $$

Step by step solution

01

Understand the associative property

To understand the associative property, consider three vectors, \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\). According to the associative property, \((\mathbf{a}+\mathbf{b}) + \mathbf{c} = \mathbf{a}+ (\mathbf{b}+\mathbf{c})\). This property signifies that the sum of vectors is the same regardless of how they are grouped.

02

Prove the sum of all five given expressions is the same

The expressions given are combinations of four vectors, \(\mathbf{v}\), \(\mathbf{w}\), \(\mathbf{x}\), and \(\mathbf{y}\), with different grouping. According to the associative property, the sum should be the same regardless of the grouping. Let's denote the sum of vectors for all five expressions as S. Hence, \(S = \mathbf{v}+ \mathbf{w}+\mathbf{x} +\mathbf{y}\).

03

Apply the associative property

Applying the associative property for S, each of the expressions given can be rewritten in the original order but with different groupings. So, whether we sum \(\mathbf{v}+\mathbf{w}\) first and then add \(\mathbf{x}\), and then \(\mathbf{y}\), or whether we sum \(\mathbf{x}+\mathbf{y}\) first and then add it to the sum of \(\mathbf{v}+\mathbf{w}\), it will result in the same vector S.

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Key Concepts

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

Associative Property

The associative property is a fundamental rule in mathematics that applies to various operations, including vector addition. In simple terms, it says that when we add multiple items, the way we group them does not affect the sum. For vectors, this is crucial.

Let's break it down using an example. Imagine you have three vectors:

• \(\mathbf{a}\)
• \(\mathbf{b}\)
• \(\mathbf{c}\)

The associative property tells us that the sum \((\mathbf{a} + \mathbf{b}) + \mathbf{c}\) is the same as \(\mathbf{a} + (\mathbf{b} + \mathbf{c})\). This means the result doesn't change whether you add \(\mathbf{a}\) and \(\mathbf{b}\) first, or \(\mathbf{b}\) and \(\mathbf{c}\) first.

This property is vital because it allows us to think of vector sums without worrying about the sequence of operations. The freedom from grouping lets us write sums without excessive parentheses. For example, we can express the sum of several vectors as \(\mathbf{v} + \mathbf{w} + \mathbf{x} + \mathbf{y}\), without needing to specify how we grouped them. It simplifies notation and calculations, making our work with vectors more straightforward.

Vectors

Vectors are essential mathematical objects that have both a direction and a magnitude, often used to describe physical quantities like force and velocity. They help us represent quantities that are not just about magnitude but also about direction.

Think of a vector like an arrow. The direction in which the arrow points represents the direction of the vector, and the length of the arrow represents its magnitude. When combining vectors, such as in vector addition, these two components—the direction and the magnitude—affect the result.

There are key operations associated with vectors:

• Addition: Combine two vectors to produce a third, resultant vector. The associative property allows the flexibility to sum multiple vectors without worrying about the order in which they are added.
• Scalar Multiplication: Multiply a vector by a number (a scalar) to change its magnitude without affecting its direction.

Understanding vectors is critical in physics and engineering, as they provide a way to analyze quantities that behave in more than one dimension. For example, when analyzing forces acting on a particle, vectors can help deduce the particle's trajectory by summing these forces, thanks to vector addition.

Proof by Induction

Proof by induction is a powerful mathematical technique, often used to prove statements about integers or sequences. It might seem a bit complex at first, but it's based on a simple idea: if something is true for the first instance and you can show it keeps being true as you progress, it's true for all instances.

Here's how it works, step by step:

• Base Case: Show that a statement holds for an initial value. This establishes that the first "domino" in our logical sequence falls.
• Inductive Step: Assume that the statement is true for some arbitrary value, say \(n\). Then demonstrate that if it's true for \(n\), it must also be true for \(n+1\). This acts like setting up a chain reaction after the first domino falls.

To connect this to vector addition, suppose we want to prove that the sum of any number of vectors obeys the associative property. We could use induction to show:

• Base Case: The associative property holds for a small number of vectors, say two or three.
• Inductive Step: If it holds for some number \(n\), it also holds for \(n+1\) vectors.

By completing these steps, we've mathematically validated that regardless of how many vectors we are working with, their sum behaves properly due to the associative property, proving it's legitimate to ignore parenthesis when adding vectors.