Vector Calculas: Simplifying expressions

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SUMMARY

The discussion focuses on simplifying the expression (a - b) · (b - c) × (c - a), which is identified as a scalar triple product. Participants clarify that the expression as written is incorrect because a scalar cannot be crossed with a vector. The correct interpretation involves rewriting the expression as (a - b) · ((b - c) × (c - a)). The original book claims the result is 0, prompting further investigation into the simplification of the cross product.

PREREQUISITES
  • Understanding of vector operations, specifically dot and cross products.
  • Familiarity with scalar triple products in vector calculus.
  • Basic knowledge of vector notation and properties.
  • Ability to interpret and manipulate mathematical expressions in LaTeX format.
NEXT STEPS
  • Review the properties of scalar triple products in vector calculus.
  • Study the derivation and simplification of cross products in vector algebra.
  • Practice writing and interpreting expressions in LaTeX for clarity in mathematical communication.
  • Explore additional resources on vector calculus, such as online tutorials or textbooks.
USEFUL FOR

Students studying vector calculus, educators teaching mathematical concepts, and anyone looking to enhance their understanding of vector operations and scalar triple products.

gotpho
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Homework Statement


(a -b) \cdot (b - c) \times (c - a)

Homework Equations





The Attempt at a Solution



Honestly I have no idea where to being. I believe this expression is a scalar triple product but I do not know how to use the properties to simplify this expression.

Sorry, kind of screw up the latex symbols. Not sure how to use it probably.
 
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Mark44 said:
As you've written it, the expression doesn't make any sense. If you calculate (a - b) \cdot (b - c), you get a scalar, which you can't cross with a vector. Do you mean
(a - b) \cdot ((b - c) X (c - a))?

If so, here's a Wikipedia article that might be of help to you - http://en.wikipedia.org/wiki/Scalar_triple_product#Scalar_triple_product

Perhaps but the book wrote it as (a - b) \cdot (b - c) X (c - a). The book says the answers is 0 but I'm clueless as to where to begin. I'm guessing we have to simplify the cross product first.
 

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