Understanding Coplanar Parallelepiped in Vector Algebra

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SUMMARY

The discussion centers on the concept of coplanar parallelepipeds in vector algebra, specifically illustrated by the equation (a × b) · c = 0, which is equivalent to the determinant det[a, b, c] = 0. In this case, the vectors a = (1,1,2), b = (2,3,4), and c = (1,-2,2) confirm that the parallelepiped formed by these vectors is coplanar. This indicates that the edges of the parallelepiped lie in the same plane, resulting in a shape that is flattened and possesses no volume.

PREREQUISITES
  • Understanding of vector operations, specifically cross product and dot product.
  • Familiarity with determinants in linear algebra.
  • Knowledge of the geometric interpretation of vectors in three-dimensional space.
  • Basic concepts of parallelepipeds and their properties.
NEXT STEPS
  • Study vector cross product and its geometric implications.
  • Learn about determinants and their role in determining coplanarity.
  • Explore the properties of parallelepipeds in vector geometry.
  • Investigate applications of coplanar vectors in physics and engineering.
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Students and professionals in mathematics, physics, and engineering who are looking to deepen their understanding of vector algebra and geometric interpretations of vectors.

cicatriz
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I have just completed a question in which I have determined:

(axb).c = 0 = det [a,b,c]

Where: a = (1,1,2) b = (2,3,4) c = (1,-2,2)


With some googling, I established this meant the parallelepiped to be coplanar but I'm not sure exactly what this means. If anyone could offer me some assistance with this I would be most grateful.


Cicatriz
 
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It means that all of the edges of your parallelepiped lie in the same plane. So the box is squashed flat and has no volume.
 
Dick said:
It means that all of the edges of your parallelepiped lie in the same plane. So the box is squashed flat and has no volume.

Thanks!
 

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