Problem 71 - Algebra and Trigonometry Real Mathematics, Real People

Question

The vector \(\mathbf{u}=\langle 1225,2445\rangle\) gives the numbers of hours worked by employees of a temp agency at two pay levels. The vector \(\mathbf{v}=\langle 12.00,10.25\rangle\) gives the hourly wage (in dollars) paid at each level, respectively. (a) Find the dot product \(\mathbf{u} \cdot \mathbf{v}\) and explain its meaning in the context of the problem. (b) Identify the vector operation used to increase wages by 2 percent.

Step-by-Step Solution

Answer

The dot product \(\mathbf{u} \cdot \mathbf{v}\) gives the total wages paid by the temp agency to its employees under the given wage rates and hours. Vector scalar multiplication is used to increase the hourly wage by 2 percent.

Step 1: Finding the dot product of the vectors

The dot product of two vectors in \(\mathbb{R}^2\), \(\mathbf{a} = \langle a_1, a_2 \rangle\) and \(\mathbf{b} = \langle b_1, b_2 \rangle\), is given by \(a_1 \cdot b_1 + a_2 \cdot b_2\). Using this definition, the dot product \(\mathbf{u} \cdot \mathbf{v}\) of the two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is calculated as:\( (1225 \cdot 12.00) + (2445 \cdot 10.25)\)

Step 2: Interpretation of the dot product in the context

In the given context, \(\mathbf{u}\) represents hours worked and \(\mathbf{v}\) represents hourly wages, so their dot product calculates the total wages paid to the employees.

Step 3: Operation to increase wages

To increase wages by 2 percent, a scalar multiplication operation is needed as we multiply the wage vector \(\mathbf{v}\) by the scalar 1.02 (representing a 2% increase)

Key Concepts

Dot Product

The dot product is a fundamental operation in vector mathematics. It is a way to multiply two vectors, resulting in a scalar. This operation combines the magnitudes of the vectors and the cosine of the angle between them when dealing with geometric vectors. However, in applications like the one in our exercise, it provides a very practical result.

Given two vectors, say \( \mathbf{u} = \langle 1225, 2445 \rangle \) and \(\mathbf{v} = \langle 12.00, 10.25 \rangle \), their dot product \( \mathbf{u} \cdot \mathbf{v} \) can be computed as:

• Multiply the respective components: \( (1225 \cdot 12.00) \) and \( (2445 \cdot 10.25) \)
• Add these products together.

When completed, this summation tells us the total dollars paid in wages. This is because the product of hours worked and hourly rate directly gives us the total earnings for each wage level, and summing these provides the total payment to all employees.

Scalar Multiplication

Scalar multiplication is another key operation in vector math. It involves multiplying a vector by a scalar (a single number) and affects each component of the vector independently.

In our context, you want to adjust the hourly wages by a specific percentage, say increasing them by 2%. Here is how scalar multiplication works:

• Identify the scalar, which in this case is \( 1.02 \) for a 2% increase.
• Multiply each component of the vector \( \mathbf{v} = \langle 12.00, 10.25 \rangle \) by this scalar: \( 12.00 \cdot 1.02 \) and \( 10.25 \cdot 1.02 \).

The result is a new vector representing the updated wages reflecting the 2% raise. This operation allows us to seamlessly adjust values across all dimensions the vector represents, such as wages in this instance.

Contextual Interpretation

Interpreting mathematical operations in the context of real-world scenarios is essential for understanding their true impact.

In our exercise, the dot product operation provides a tangible outcome: the total wages paid to employees. Here's why this interpretation is helpful:

• The dot product brings together hours worked and rates, key inputs in wage calculation, to compute total wages.
• It simplifies the understanding of payroll costs by consolidating multiple wage levels into a single figure.

Similarly, the scalar multiplication of the wage vector increases each component uniformly by a set percentage. This operation helps in:

• Quickly updating salary information due to increments or cuts.
• Maintaining consistency across wage adjustments, ensuring equal percentage changes.

These interpretations make vector operations practical, connecting mathematical computations to context-specific decision-making processes and outcomes.