9464

Dot and Cross Products Related to Complex Multiplication

A complex number can be considered as a vector and vice versa, both points of view having their own context. The operations transforming vectors and complex numbers are particular to them; vectors use the dot and cross products while complex numbers use multiplication and conjugation (written using an overbar). How are these two pairs of operations related to one another when their operands are identified either as vectors or as complex numbers?
This Demonstration shows the equation relating the two kinds of operations (writing complex numbers in uppercase letters, and , and their corresponding vectors as and ).
Some possible questions to address are:
When are and perpendicular? (answer: when );
When are and parallel? (answer: when —the usual test cannot be used if );
More generally, what is the value of the area of the parallelogram with sides and ? (answer: ).
When is ? (answer: when is a real number).
Prove that when , then .
Prove that the triangle with side lengths , , and is a right triangle.
Show that the magnitude of the projection of onto (the length of the side of the red triangle that is parallel to ) is equal to .

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+