Problem 60 - 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? $$(c+d) \mathbf{u}=c \mathbf{u}+d \mathbf{u}$$

Step-by-Step Solution

Answer

The given expression \((c+d) \mathbf{u}=c \mathbf{u}+d \mathbf{u}\) holds true, proving the distributive property for vectors is valid.

Step 1: Define and Identify the Variables

Let's define the scalar quantities be c and d, and vector u as \(u=a_1 i + b_1 j \). Now let's identify each part of the equation. On the left side, we have \((c+d)u\), which means we're adding the scalars before multiplying by the vector, and on the right side, we have \(cu + du\), which means we're multiplying each scalar by the vector and then adding the results.

Step 2: Perform Operations on Left side of the Equation

We multiply the scalar sum (c+d) with the vector u to get: \((c+d)u = (c+d)(a_1 i + b_1 j) = (c+d)a_1 i + (c+d)b_1 j\).

Step 3: Perform Operations on Right side of the Equation

We add the distributive operations \(cu + du = (ca_1 + da_1)i + (cb_1 + db_1)j\).

Step 4: Compare the Results

By comparing the results from both sides of the equation, we find that they are identical, thus proving that the distributive property of vectors is valid.

Key Concepts

Distributive Property

The distributive property in vector algebra is similar to its arithmetic counterpart. It describes how scalar multiplication distributes over vector addition, indicating that the order of operations doesn't affect the result. In vectors, this property can be expressed as

• For scalars and vectors: \[(c+d) \mathbf{u} = c \mathbf{u} + d \mathbf{u}\] This equation shows that adding scalars before multiplying them with a vector results in the same vector as multiplying each scalar separately by the vector and then adding those vectors.

Applying this to the given exercise, both the left side and right side of the equation resolve to:

• Left side: \[(c+d)\mathbf{u} = (c+d)(a_1 \mathbf{i} + b_1 \mathbf{j}) = (c+d)a_1 \mathbf{i} + (c+d)b_1 \mathbf{j}\]
• Right side: \[(c \mathbf{u} + d \mathbf{u}) = c(a_1 \mathbf{i} + b_1 \mathbf{j}) + d(a_1 \mathbf{i} + b_1 \mathbf{j}) = (ca_1 + da_1) \mathbf{i} + (cb_1 + db_1) \mathbf{j}\]

Both sides simplify to the same vector, reaffirming the distributive property in scalar multiplication with vectors.

Associative Property

The associative property allows us to regroup vectors during addition without changing the result. This property states that no matter how we group the vectors, the sum remains the same:

• \[(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})\]

This property is crucial because it simplifies the process of adding multiple vectors by allowing us to combine them in any order.
For example, if we have vectors \(\mathbf{u}, \mathbf{v}, \text{ and } \mathbf{w},\) and calculate their resultant as \((\mathbf{u} + \mathbf{v}) + \mathbf{w},\) it will be the same as \(\mathbf{u} + (\mathbf{v} + \mathbf{w}).\)This flexibility makes vector computations more intuitive and manageable, especially in larger systems.

Commutative Property

The commutative property in vector addition tells us that the order in which vectors are added does not matter. It is defined as:

• \[\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}\]

Regardless of the order, the resultant vector is the same. This property simplifies and offers flexibility in calculations since rearranging vectors doesn't affect the final result.
Understanding this property is essential in vector algebra, as it facilitates calculations and proofs by providing more pathways to approach problems.
In practice, you can rearrange terms anytime you are adding vectors without changing the sum, which is especially useful for simplifying complex vector expressions.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a real number), scaling its magnitude without changing its direction. For a vector \(\mathbf{u} = a_1 \mathbf{i} + b_1 \mathbf{j},\) the operation looks like this:

• \[c \mathbf{u} = c(a_1 \mathbf{i} + b_1 \mathbf{j}) = (ca_1) \mathbf{i} + (cb_1) \mathbf{j}\]

A key aspect of scalar multiplication is that it distributes over vector addition and subtraction, as shown in the distributive property.
It’s important in physics and engineering to depict forces, velocities, and other physical quantities where one factor (like the magnitude) needs to be altered.
After scalar multiplication, the vector’s direction stays the same unless the scalar is negative, in which case the vector points in the opposite direction.

Vector Addition

Vector addition combines two or more vectors to create a resultant vector. The process involves adding corresponding components of the vectors. For vectors \(\mathbf{u} = a_1 \mathbf{i} + b_1 \mathbf{j}\) and \(\mathbf{v} = a_2 \mathbf{i} + b_2 \mathbf{j},\) the resultant \(\mathbf{r}\) is given by:

• \[\mathbf{r} = \mathbf{u} + \mathbf{v} = (a_1 + a_2) \mathbf{i} + (b_1 + b_2) \mathbf{j}\]

This straightforward component-wise addition provides a geometric approach, represented by the parallelogram method or triangle rule.
Vector addition is crucial in physics and engineering to model and understand combined forces, motions, and other vector quantities.Consider practice with such operations to become comfortable working with vectors in different dimensions and scenarios.