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


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.


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


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 `vecu``vecv=||vecu||*||vecv||*costheta` where θ is the acute angle between vectors u and v. Note `vecu` and `vecv` are perpendicular IFF `vecu``vecv = 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

James Sousa's MathIsPower4U - Calc II:
Dot Product of Vectors (2D)
Dot Product of Vectors from a Graph (2D)
Dot Product of Vectors (3D)
Find a Component of a Vector so Two Vectors are Orthogonal (2D)
Find a Component of a Vector so Two Vectors are Orthogonal (3D)
Determining the Angle Between Two Vectors (2D)
Find the Angle Between Two Vectors (3D)