Understanding Coplanar Parallelepiped in Vector Algebra

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In summary, the conversation discussed the calculation of (axb).c and the meaning of the resulting value, which is det [a,b,c] = 0. This indicates that the parallelepiped formed by the vectors a, b, and c is coplanar, meaning that all of its edges lie in the same plane and it has no volume.
  • #1
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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  • #2
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.
 
  • #3
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!
 

What is a coplanar parallelepiped?

A coplanar parallelepiped is a three-dimensional shape that has six parallelogram faces. This means that all the faces of the shape lie on the same plane.

How is a coplanar parallelepiped different from a regular parallelepiped?

A regular parallelepiped, also known as a rectangular prism, has rectangular faces while a coplanar parallelepiped has parallelogram faces. Additionally, all the faces of a coplanar parallelepiped lie on the same plane, while a regular parallelepiped may not.

What are some properties of a coplanar parallelepiped?

Some properties of a coplanar parallelepiped include having six faces, twelve edges, and eight vertices. It also has opposite faces that are parallel and equal in size and opposite edges that are equal in length. The sum of the angles between any two adjacent faces is always 180 degrees.

How can you calculate the volume of a coplanar parallelepiped?

The volume of a coplanar parallelepiped can be calculated by multiplying the length of its base by its height. The base and height must be perpendicular to each other, and the height must be measured as the shortest distance between the base and its opposite face. The formula for volume is V = Bh, where B is the area of the base and h is the height.

What are some real-life examples of coplanar parallelepipeds?

Coplanar parallelepipeds can be found in various structures and objects, such as buildings, boxes, and shelves. They are also commonly used in engineering and architecture, as well as in everyday objects like books, tablets, and cereal boxes.

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