Problem 58 - 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? $$(\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})$$

Step-by-Step Solution

Answer

The result proves that vector addition is associative, i.e., \( (\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w}) \).

Step 1: Add vectors \( \mathbf{u} \) and \( \mathbf{v} \) then add the result to \( \mathbf{w} \)

Starting with \( (\mathbf{u}+\mathbf{v})+\mathbf{w} \), the addition of vectors \( \mathbf{u} \) and \( \mathbf{v} \) results in a new vector \( \mathbf{x} \) which components are \( x_{i}=a_{1}+a_{2} \) and \( x_{j}=b_{1}+b_{2} \). Adding this vector \( \mathbf{x} \) to \( \mathbf{w} \) results in a vector \( \mathbf{y} \) with \( y_{i}=x_{i}+a_{3} \) and \( y_{j}=x_{j}+b_{3} \). Substitute the values for \( x_{i} \) and \( x_{j} \) back into these equations to derive that \( y_{i}=a_{1}+a_{2}+a_{3} \) and \( y_{j}=b_{1}+b_{2}+b_{3} \).

Step 2: Add vectors \( \mathbf{v} \) and \( \mathbf{w} \) then add the result to \( \mathbf{u} \)

Now looking at \( \mathbf{u}+(\mathbf{v}+\mathbf{w}) \), the addition of vectors \( \mathbf{v} \) and \( \mathbf{w} \) results in a new vector \( \mathbf{z} \) which components are \( z_{i}=a_{2}+a_{3} \) and \( z_{j}=b_{2}+b_{3} \). Adding this vector \( \mathbf{z} \) to \( \mathbf{u} \) results in a vector \( \mathbf{p} \) with \( p_{i}=a_{1}+z_{i} \) and \( p_{j}=b_{1}+z_{j} \). Substitute the values for \( z_{i} \) and \( z_{j} \) back into these equations to derive that \( p_{i}=a_{1}+a_{2}+a_{3} \) and \( p_{j}=b_{1}+b_{2}+b_{3} \).

Step 3: Compare the results

Given that \( y_{i}=p_{i} \) and \( y_{j}=p_{j} \), it can be concluded that \( (\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w}) \). So the property proven is associative.

Key Concepts

Associative Property of Vectors

The associative property is a fundamental principle in vector algebra, relating to how vectors can be grouped during addition. Simply put, it suggests that no matter how we pair vectors together in an addition process, the resulting vector will be the same.

To see this in action, consider three vectors, labeled as \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\), each with their own components along the i (horizontal) and j (vertical) axes. According to the associative property, adding these vectors together in any grouped manner should lead to the same sum. So, \((\mathbf{u}+\mathbf{v})+\mathbf{w}\) will be identical to \(\mathbf{u}+(\mathbf{v}+\mathbf{w})\).

The textbook solution validates this property by breaking down the addition into steps and comparing the final components of the resulting vectors. The exercise demonstrates that by grouping and then adding the components of the vectors, the order of the grouping doesn’t affect the final components, which stands as proof of the associative property in vectors.

Vector Addition

Vector addition involves combining the corresponding components of two or more vectors to produce a new vector. It operates under the rule that each component of the vectors being added together, that is, their influence in a specific dimension, summatively contributes to the corresponding component of the resulting vector.

In the context of our vectors \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\), we identify their respective components as \(a_{1}\), \(a_{2}\), \(a_{3}\) on the i-axis and \(b_{1}\), \(b_{2}\), \(b_{3}\) on the j-axis. When added together, their i-components and j-components are summed separately to create the components of the new vector.

This method ensures a straightforward and consistent way to combine vectors, which is essential for analysis in physics, engineering, and many other fields where vectors are routinely used to represent quantities with both magnitude and direction.

Vector Components

Vectors are often described by their components, which express the influence of the vector in particular directions. Each component can be seen as a projection of the vector onto an axis of a coordinate system. For example, in a 2D coordinate system, a vector \(\mathbf{v}\) is commonly represented by its components in the horizontal (i) and vertical (j) directions as \(\mathbf{v} = a_{i}\mathbf{i} + b_{j}\mathbf{j}\).

These components, \(a_{i}\) and \(b_{j}\), are essentially the 'building blocks' of the vector and determine its overall direction and magnitude. Understanding vector components is crucial for more complex operations in vector calculus and physics, such as expressing forces, velocities, and other vector quantities in a clear and manageable form.