Vector Value Analysis Worksheet Vector Value Analysis
Application: Dot Products in Business Cost Analysis
Student:
Date:
Warm-Up: Dot Product Refresh
Calculate the dot product for the following vector pairs. Show each step of your calculation.
1. \(\vec{u} = \langle 5, -2 \rangle, \vec{v} = \langle 3, 4 \rangle\)
2. \(\vec{a} = \langle 1, 0, 4 \rangle, \vec{b} = \langle -2, 7, 3 \rangle\)
Case Study: Mia's Catering
During the video segment (2:32–4:15), record the data components for the catering order as vectors. Use the values provided by the narrator.
Quantity Vector \(\vec{q}\)
\(\langle\)
,
,
,
\(\rangle\)
Price Vector \(\vec{p}\)
\(\langle\)
,
,
,
\(\rangle\)
Calculation & Interpretation
Show the calculation for \(\vec{q} \cdot \vec{p}\) and explain what the numerical result means for the business.
The School Store Activity
Step 1: Inventory Selection
Identify 5 different items you want to sell in your school store. Assign a quantity (how many you have) and a price (how much each costs) to each item.
Item Name Quantity (Units) Unit Price ($) e.g., Pencil
Your Store Vectors
Inventory \(\vec{i} = \langle \_\_, \_\_, \_\_, \_\_, \_\_ \rangle\)
Price \(\vec{p} = \langle \_\_, \_\_, \_\_, \_\_, \_\_ \rangle\)
Step 2: Partner Audit
Swap papers with a classmate. Use their data vectors to find the total value of their inventory.
Partner Name
Partner's Vectors
Record vectors here...
Calculation (\(\vec{i} \cdot \vec{p}\)):
Total Store Value:
$ _________
Final Reflection
1. Scalability & Automation
Imagine a company with 100,000 unique products. Why is the "vectorized" approach (dot product) more powerful for a programmer or data analyst than traditional row-by-row multiplication?
2. Beyond Dollars & Cents
If we added a third vector \(\vec{w}\) representing the weight of each item, how could a shipping company use a dot product to calculate the total weight of a truck's cargo?
Vector Value Analysis Lesson Plan Vector Value Analysis
Lesson Facilitation Guide | 11th Grade Pre-Calculus
Duration
45 Minutes
Objective
Students will apply the vector dot product to solve real-world cost analysis problems, interpreting the resulting scalar value within a business context.
Standards
HSVM.B.4: Multiply a vector by a scalar.HSVM.B.5: Solve problems involving velocity and other quantities that can be represented by vectors.MP.4: Model with mathematics.
Materials Needed
Student Worksheets
Video (Link provided)
Scientific Calculators
Classroom Timer
Instructional Sequence
0:00 - 0:05 | Warm-Up: Skill Check
5 MIN
Reviewing the mechanical process of dot products. Monitor students for component-wise multiplication and summation errors.
0:05 - 0:15 | Video Case Study
10 MIN
Watch Mia's Catering segment (2:32–4:15). Discuss: How do vectors act as "data containers"? Point out that vectors don't just store spatial direction, but lists of related numerical data.
URL: https://www.youtube.com/watch?v=VFSnidBsbss
0:15 - 0:35 | The School Store Activity
20 MIN
Pairs create an inventory list and price list. Encourage variety in quantities and prices. The Switch: Students swap papers. This ensures they practice calculating using someone else's data structure, simulating an "audit."
0:35 - 0:45 | Reflection & Share Out
10 MIN
Focus on efficiency. Connect this to Computer Science: dot products are a core operation in data processing and machine learning.
Worksheet Answer Key
Warm-Up Solutions
1. \((5 \cdot 3) + (-2 \cdot 4) = 15 - 8 = \mathbf{7}\)
2. \((1 \cdot -2) + (0 \cdot 7) + (4 \cdot 3) = -2 + 0 + 12 = \mathbf{10}\)
Mia's Catering Data
Quantity Vector \(\vec{q}\): \(\langle 90, 30, 25, 60 \rangle\)
Price Vector \(\vec{p}\): \(\langle 2.50, 15.00, 6.00, 1.50 \rangle\)
Calculation: \((90 \cdot 2.5) + (30 \cdot 15) + (25 \cdot 6) + (60 \cdot 1.5) = 225 + 450 + 150 + 90 = \mathbf{915}\)
Vector Value Analysis Slides Pre-Calculus
Vector Value
Analysis
Applying the dot product to real-world business scenarios.
Warm-Up: Dot Product
Scenario A
\(\vec{u} = \langle 5, -2 \rangle\)
Calculate: \(\vec{u} \cdot \vec{v}\)
Scenario B
\(\vec{a} = \langle 1, 0, 4 \rangle\)
Calculate: \(\vec{a} \cdot \vec{b}\)
Case Study: Mia's Catering
Embedded media
Observation Guide
What are the dimensions of the vectors?
What do the components represent?
How is the result useful?
Activity: The School Store
1
Setup Shop
Choose 5 items for your inventory. Set the quantity and price for each.
2
Vectorize
Convert your table into two vectors:
\(\vec{i}\) (Quantity)
3
The Audit
Swap with a partner. Calculate the total value of their store using the dot product.
Closing Thought
"Why is the dot product better for large-scale business than calculating totals item-by-item?"
Scalability
Handle millions of data points simultaneously.
Dimensionality
Easily add more 'dimensions' (tax, weight, labor).
