Problem 52 - Algebra and Trigonometry

Question

The components of \(\mathbf{v}=180 \mathbf{i}+450 \mathbf{j}\) represent the respective number of one-day and three-day videos rented from a video store. The components of \(\mathbf{w}=3 \mathbf{i}+2 \mathbf{j}\) represent the prices to rent the one-day and three-day videos, respectively. Find \(\mathbf{v} \cdot \mathbf{w}\) and describe what the answer means in practical terms.

Step-by-Step Solution

Answer

The dot product, \(\mathbf{v} \cdot \mathbf{w}\), equals 1440, which represents the total revenue in dollars generated by the video store from renting one-day and three-day videos.

Step 1: Understanding the Dot Product

The dot product of two vectors \(\mathbf{v} = a\mathbf{i} + b\mathbf{j}\) and \(\mathbf{w} = c\mathbf{i} + d\mathbf{j}\) is given by \(\mathbf{v} \cdot \mathbf{w} = ac + bd\). It multiplies corresponding components of the vectors and sums the results.

Step 2: Calculation of the dot product

To find the dot product of \(\mathbf{v} = 180\mathbf{i} + 450\mathbf{j}\) and \(\mathbf{w} = 3\mathbf{i} + 2\mathbf{j}\), multiply the corresponding components and add them: \(\mathbf{v} \cdot \mathbf{w} = (180*3) + (450*2) = 540 + 900 = 1440.\)

Step 3: Interpretation of the Result

The result of the dot product, 1440, is a scalar quantity. Given the context of the problem, this represents the total revenue generated by the video store from the renting of one-day and three-day videos.