Find the volume of the tetrahedron with the given vertices. (1,1,1),(0,0,0),(2,1,-1),(-1,1,2)

Name: Problem 42, Chapter 3: Elementary Linear Algebra
Uploaded: 2022-02-18T04:42:59-08:00
Duration: 3 min 48 s
Channel: Haricharan Gupta
Description: Find the volume of the tetrahedron with the given vertices. (1,1,1),(0,0,0),(2,1,-1),(-1,1,2)

Find the volume of the tetrahedron with the given vertices....

Key Concepts

Vectors

Vectors are mathematical objects that have both magnitude and direction and are used to represent positions, displacements, or directions in space. They form the basis for constructing edges of geometric figures and are crucial in calculations involving spatial relationships.

Scalar Triple Product

The scalar triple product of three vectors is a method for determining the volume of the parallelepiped defined by those vectors. It is computed as the dot product of one vector with the cross product of the other two, and its absolute value gives the volume of the parallelepiped.

Determinant

The determinant is a scalar value associated with a square matrix that provides important information about the linear transformation represented by the matrix. In geometry, the determinant is used to compute areas, volumes, and to test for linear independence of vectors.

Volume of a Tetrahedron

The volume of a tetrahedron can be derived from the volume of a parallelepiped formed by three vectors from one vertex, by taking one sixth of that volume. This approach leverages the scalar triple product to obtain the desired measurement.

Transcript

00:01 As well as students, we are given a tetraedron and its vertices are given as depth.

00:04 As well, we need to find the volume of the tetrodron.

00:07 We can say that volume is nothing but 1 divided by 6 times of, okay? and the determinant of those terms, those vertices, how we will write them in a matrix form? so first column can be written as x1, x2, x3, x4, so basically 1, 0, 2 and minus 1, okay? second column is nothing but y1, y2, y1, sorry, y4, so basically 1 here is 0 and it is nothing but 1, again one okay now the third column can be written as z1 z2 z3 z4 so basically one it will be zero near minus 1 and 2 and the fourth column can be written as 1 1 1 1 it's all about the basic expression now we will expand it by cofactors 1 by 6 times of okay so how we will write see we just make a column it is not a matrix format it is a column bracket okay don't need to confuse so basically we will write here one times of okay students and one times of 0 0 0 -01 here 1 minus 1 1 1 1 2 1 okay after that minus and here it this one times of we will have to skip this row and this column what we will get 0 to minus 1 here 0 minus 1 2 1 1 okay now plus this 1 okay times of we will have to skip this row and this column what we will 0 0 0 and here 2 1 minus 1 1 1 okay 0 2 1 and 1 minus 1 1 1 okay now minus of what we can do here 1 times of okay students so basically 1 times of we have to skip this row in this column what we'll get 0 0 0 0 0 2 1 minus 1 and minus 1 1 2 now this is whole bracket now we need to solve this whole term by calculation 1 by 6 times so we can write it see one no need to write and here it will be zero minus zero after that plus one times.

02:05 Okay, one times of two is nothing but two minus one.

02:07 One times of minus one is nothing but minus one...

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