Problem 49
Question
Suppose that \(\mathbf{u}=\langle 2000,5000\rangle\) represents the number of units of battery \(A\) and \(B\), respectively, produced by a company and \(\mathbf{v}=\langle 8.40,6.50\rangle\) represents the price (in dollars) of a 10 -pack of battery \(A\) and \(B\), respectively. Compute and interpret \(\mathbf{u} \cdot \mathbf{v}\).
Step-by-Step Solution
Verified Answer
The dot product \( \mathbf{u} \cdot \mathbf{v} = 49300 \) represents the total revenue from sales of batteries A and B.
1Step 1: Understand Dot Product
The dot product (also known as the scalar product) of two vectors \( \mathbf{u} \) and \( \mathbf{v} \) is calculated by multiplying corresponding components and then summing those products. If \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \), then the dot product \( \mathbf{u} \cdot \mathbf{v} \) is given by \( u_1 \cdot v_1 + u_2 \cdot v_2 \).
2Step 2: Substitute Values
Substitute the given values into the dot product formula: \( \mathbf{u} = \langle 2000, 5000 \rangle \) and \( \mathbf{v} = \langle 8.40, 6.50 \rangle \). Compute \( 2000 \times 8.40 \) and \( 5000 \times 6.50 \).
3Step 3: Calculate Products
Calculate the individual products: \( 2000 \times 8.40 = 16800 \) and \( 5000 \times 6.50 = 32500 \).
4Step 4: Sum the Products
Add the results from the previous step: \( 16800 + 32500 = 49300 \).
5Step 5: Interpret the Result
The result \( 49300 \) represents the total revenue (in dollars) from the sale of 2000 units of battery \( A \) and 5000 units of battery \( B \), reflecting the combined contribution of both battery types to the company's total revenue for the given production quantities and selling prices.
Key Concepts
VectorsScalar MultiplicationRevenue Calculation
Vectors
Vectors are mathematical objects that have both magnitude and direction. They are often used to represent quantities in physics and engineering, such as displacement, velocity, and, as in the given exercise, quantities in economics.
A vector can be represented as an ordered pair or tuple, like \( \mathbf{u} = \langle 2000, 5000 \rangle \). This notation indicates two components: in this case, the production quantities of battery types \( A \) and \( B \). Understanding how vectors can structurally represent multi-component data is fundamental in grasping more complex operations.
Vectors can be manipulated through operations such as addition, subtraction, and, significantly for this exercise, the dot product. The dot product itself turns a vector’s information into a single, scalar value, which is great for interpreting relationships like total revenue.
A vector can be represented as an ordered pair or tuple, like \( \mathbf{u} = \langle 2000, 5000 \rangle \). This notation indicates two components: in this case, the production quantities of battery types \( A \) and \( B \). Understanding how vectors can structurally represent multi-component data is fundamental in grasping more complex operations.
Vectors can be manipulated through operations such as addition, subtraction, and, significantly for this exercise, the dot product. The dot product itself turns a vector’s information into a single, scalar value, which is great for interpreting relationships like total revenue.
Scalar Multiplication
Scalar multiplication involves multiplying a vector by a scalar (a single number) which results in a vector being scaled in magnitude but not direction.
In the context of the dot product, while the term 'scalar multiplication' might initially imply the same operation, it actually refers to the component-wise multiplication involved.
For example, given vectors \( \mathbf{u} = \langle 2000, 5000 \rangle \) and \( \mathbf{v} = \langle 8.40, 6.50 \rangle \), this becomes clear in the step of calculating \( 2000 \times 8.40 \) and \( 5000 \times 6.50 \). This performs scalar multiplication for each component of \( \mathbf{u} \) with the corresponding component of \( \mathbf{v} \). Summing these results gives a scalar value, which, unlike a vector, has magnitude but no direction.
In the context of the dot product, while the term 'scalar multiplication' might initially imply the same operation, it actually refers to the component-wise multiplication involved.
For example, given vectors \( \mathbf{u} = \langle 2000, 5000 \rangle \) and \( \mathbf{v} = \langle 8.40, 6.50 \rangle \), this becomes clear in the step of calculating \( 2000 \times 8.40 \) and \( 5000 \times 6.50 \). This performs scalar multiplication for each component of \( \mathbf{u} \) with the corresponding component of \( \mathbf{v} \). Summing these results gives a scalar value, which, unlike a vector, has magnitude but no direction.
Revenue Calculation
Revenue calculation in this context uses the concept of the dot product to determine total revenue from multiple product lines.
The exercise computes \( \mathbf{u} \cdot \mathbf{v} \), which gives a single number: \( 49300 \). This value represents the total revenue when you produce and sell 2000 units of Battery A at \( 8.40 \) dollars each and 5000 units of Battery B at \( 6.50 \) dollars each.
This approach efficiently combines production and pricing information into a singular understanding of the company's sales performance. It eliminates the need for tedious repetitive calculations by leveraging vector mathematics to summarily deduce comprehensive insights from data.
The exercise computes \( \mathbf{u} \cdot \mathbf{v} \), which gives a single number: \( 49300 \). This value represents the total revenue when you produce and sell 2000 units of Battery A at \( 8.40 \) dollars each and 5000 units of Battery B at \( 6.50 \) dollars each.
This approach efficiently combines production and pricing information into a singular understanding of the company's sales performance. It eliminates the need for tedious repetitive calculations by leveraging vector mathematics to summarily deduce comprehensive insights from data.
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