Why is the Vector (Cross) Product pxq Perpendicular to a Plane?

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SUMMARY

The vector (cross) product of two vectors p and q is always perpendicular to the plane defined by these vectors. This property is inherent in the definition of the cross product, which can be explored through resources on multivariable calculus. The discussion references two online introductions that elaborate on the algebraic expression of the cross product and its geometric implications. Understanding this concept is essential for applications in physics and engineering where vector operations are fundamental.

PREREQUISITES
  • Understanding of vector operations, specifically the cross product
  • Familiarity with multivariable calculus concepts
  • Basic knowledge of geometric interpretations of vectors
  • Ability to analyze algebraic expressions related to vectors
NEXT STEPS
  • Explore the properties of the cross product in depth
  • Study the geometric interpretations of vector operations
  • Learn about applications of the cross product in physics, such as torque and angular momentum
  • Review multivariable calculus resources, particularly chapters on vector calculus
USEFUL FOR

Students of mathematics and physics, educators teaching vector calculus, and professionals in engineering fields who require a solid understanding of vector operations and their applications.

matqkks
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Given two vectors p and q, why is the vector (cross) product pxq perpendicular to the plane containing these vectors?
Is there a geometric or physical way of explaining why?
Are there any real life examples we can draw from?
 
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Well, what is your definition of cross product? Sometimes this (perpendicularity) is part of the definition.
 
Two links posted recently in these forums are online introductions to multivariable calculus which each contain a chapter relating the algebraic expression for the components of a cross product to the property of being perpendicular to its factors:

http://synechism.org/drupal/cfsv/
http://www.owlnet.rice.edu/~fjones/

Greg Egan used to have a nice demonstration of the correspondence between geometric and componentwise definitions of the dot product (which is taken as the starting point by the links above), but I can't seem to find it now. Never mind, Google knows lots more. You can shop around for the one that makes most sense to you.
 

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