Vector Triple Product: Are a & c Parallel or Collinear?

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SUMMARY

The discussion centers on the relationship between vectors a, b, and c in the context of the vector triple product. It establishes that while the equation (a × b) × c = a × (b × c) generally does not hold, it can be true under specific conditions, leading to the conclusion that either vector b is parallel to (a × c) or vectors a and c are collinear. The confusion arises regarding the terminology, with the participant questioning whether a and c can be considered parallel instead of collinear, despite references in literature favoring the term collinear. The consensus is that in three-dimensional space, parallel vectors are indeed collinear, as one vector is a scalar multiple of the other.

PREREQUISITES
  • Understanding of vector operations, specifically cross products.
  • Familiarity with the concepts of parallel and collinear vectors.
  • Basic knowledge of three-dimensional coordinate systems.
  • Experience with vector algebra and properties in linear algebra.
NEXT STEPS
  • Study the properties of vector cross products in detail.
  • Explore the implications of vector collinearity and parallelism in three-dimensional space.
  • Learn about the geometric interpretations of vector operations.
  • Investigate advanced topics in linear algebra, such as vector spaces and transformations.
USEFUL FOR

Students and professionals in mathematics, physics, and engineering who are dealing with vector analysis, particularly those interested in understanding the nuances of vector relationships in three-dimensional space.

debjit625
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Hi all got a confusion
In many books I saw , authors used a specific statement here is it

a,b,c are vectors and axb is (" a cross b")
In general
(axb)xc ≠ ax(bxc)
but if
(axb)xc = ax(bxc)
solving it we get
bx(axc)=0
then it implies
either b is parallel to (axc)
or a and c are collinear.

Now my question is can I say a and c are parallel rather co linear ,my confusion arise as all books I referred they all say its co linear.

Now I think in general a and c doesn't have to lie in a same line to get the specific definition of co linearity ,
but I am not sure.

Thanks
 
Physics news on Phys.org
In 3D we represent a vector with three numbers (x,y,z)
All vectors 'start' in (0,0,0), so if they are parallel they are also colinear (one is a multiple of the other).
 

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