- Calculate the dot product of two given vectors
- Find angles between vectors and determine when vectors are perpendicular using dot products
- Explain what is meant by the vector projection of one vector onto another vector, and describe how to compute it
- Calculate the work done by a given force
Abstract Vector Applications
Vectors aren’t limited to physical quantities like force and position. They’re powerful tools for organizing and processing data.
Consider a fruit vendor who sells apples, bananas, and oranges. He can represent his daily sales with a quantity vector [latex]\mathbf{q} = \langle 30, 12, 18 \rangle[/latex] ([latex]30[/latex] apples, [latex]12[/latex] bananas, [latex]18[/latex] oranges) and his prices with a price vector [latex]\mathbf{p} = \langle 0.50, 0.25, 1.00 \rangle[/latex].
The dot product [latex]\mathbf{q} \cdot \mathbf{p} = 30(0.50) + 12(0.25) + 18(1.00) = 15 + 3 + 18 = 36[/latex] gives his total daily revenue: [latex]$36[/latex].
This approach extends to any number of dimensions. If the vendor adds grapefruit, he simply uses four-dimensional vectors. The dot product calculation follows the same pattern: multiply corresponding components and add the results.
AAA Party Supply Store sells invitations, party favors, decorations, and food service items such as paper plates and napkins. When AAA buys its inventory, it pays [latex]25¢[/latex] per package for invitations and party favors. Decorations cost AAA [latex]50¢[/latex] each, and food service items cost [latex]20¢[/latex] per package. AAA sells invitations for [latex]$2.50[/latex] per package and party favors for [latex]$1.50[/latex] per package. Decorations sell for [latex]$4.50[/latex] each and food service items for [latex]$1.25[/latex] per package.
During the month of May, AAA Party Supply Store sells [latex]1258[/latex] invitations, [latex]342[/latex] party favors, [latex]2426[/latex] decorations, and [latex]1354[/latex] food service items. Use vectors and dot products to calculate how much money AAA made in sales during the month of May. How much did the store make in profit?
[latex]\\[/latex]
If AAA sells [latex]1408[/latex] invitations, [latex]147[/latex] party favors, [latex]2112[/latex] decorations, and [latex]1894[/latex] food service items in the month of June, use vectors and dot products to calculate their total sales and profit for June.