Volume of Parallelopiped and Tetrahedron

A parallelopiped is a 3-D object, each of whose faces is a parallelogram.

A rectangular parallelopiped is the one, whose faces are rectangular.

A cube is a rectangular parallelopiped, whose edges are of equal length.

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A tetrahedron is a 3-D object, whose all faces are triangular. It can be shown that a parallelopiped can be decomposed into 6 tetrahedra.

Image Source : https://www.dune-project.org/

For a visual proof :  1 Parallelopipded  = 6 Tetrahedra


 

Let there be 2 such objects, whose coterminus edges are identical. Then, the volume of the tetrahedron is {\dfrac 1 6}th of that of the parallelopiped. The expression can also be obtained using the following theorem :

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Proof of Section Formula using Vectors

The section formula is one of the important results in the geometry. We will obtain the section formula for internal as well as external division, using vectors.


Section Formula for Internal Division

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Section Formula for External Division

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Vector Algebra : Theorems (I)

In this blog, we will prove fundamental theorems in vector algebra. These theorems are in accordance with the +2 curriculum.

Theorem 1 : Collinear Vectors

vectors_theorems01-00.jpgTheorem 2 : Non-collinear and Coplanar Vectors

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Theorem 3 : Coplanar Vectors

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Theorem 4 : Non-coplanar Vectors

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