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Calculus III

Course: Calculus III
Topic: Vectors and Geometry of Space
Subtopic: Dot Product

Overview

A dot product is one method of multiplying two vectors. The operation is written with a big dot between the vectors such as <1,2,3>•<4,5,6>. The answer to a dot product is a number (the answer to this example is 32), thus it is also known as a scalar product. (Don't confuse this with the product of a vector and a scalar! E.g., 3<4,5,6>=<12,15,18>.) This dot product has a geometric interpretation related to the two vectors and is used in a variety of applications of vectors. The dot product is also related to the angle between the two vectors in 2D or in 3D so is often used to find that angle.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 13.3.1 Find the dot product of two vectors using either formula in both two and three dimensions
• 13.3.2 Find the angle between two vectors via dot product formula
• 13.3.3 Determine if two vectors are orthogonal via dot product
• 13.3.4 Understand a geometric interpretation of dot product
• 13.3.5 Use dot product to perform vector analysis particularly with real-life applications in STEM fields including calculating work, components of a force, parallel forces, and normal forces

Terminology

Terms you should be able to define: dot product, scalar product, orthogonal, work, force, components of a force, parallel forces, normal forces

Formulae to have in your notes: Both methods of finding the dot product which are <<a,b,c>><<d,e,f>>=ad+be+cf and vecuvecv=||vecu||*||vecv||*costheta where θ is the acute angle between vectors u and v. Note vecu and vecv are perpendicular IFF vecuvecv = 0.

Supplemental Resources (recommended)

Explore the geometric interpretation of dot product via one of the following interactive sites:

1. Explore CalcPlot3D's Dot Product Exploration. (In CalcPlot3D: Menu >> File >> View Vector Explorations >> in left sidebar choose "Dot Product Exploration" from the drop down box. Play around by pulling the red and blue vectors on the 2D graph to see how the dot product changes.)
2. Go to Math Insight's Dot Product article. It's a pretty good read all around but in particular scroll down and play with the GeoGebra applet.
3. Go to Falstad's Dot Product Applet is one last option for play if you need it.

Supplemental Resources (optional)

Better Explained's article Understanding the Dot Product

Tevian Dray's paper The Geometry of Dot and Cross Products

Paul's OL Notes - Calc II: Dot Product

Patrick JMT Just Math Tutorials: Dot Product