Calculation Of Area And Volume

Prove that the volume of a tetrahedron with coterminous edges

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%5Ctext%7BVolume%7D%20%3D%20%5Cfrac%7B1%7D%7B6%7D%20%5Cleft%7C%20%5Coverrightarrow%7Ba%7D%20%5Ccdot%20(%5Coverrightarrow%7Bb%7D%20%5Ctimes%20%5Coverrightarrow%7Bc%7D)%20%5Cright%7C

Hence, find the volume of the tetrahedron whose coterminous edges are

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Step 1: The formula for the volume of a tetrahedron with coterminous edges

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is given by:

V%20%3D%20%5Cfrac%7B1%7D%7B6%7D%20%5Cleft%7C%20%5Coverrightarrow%7Ba%7D%20%5Ccdot%20(%5Coverrightarrow%7Bb%7D%20%5Ctimes%20%5Coverrightarrow%7Bc%7D)%20%5Cright%7C

Step 2: First, we compute the cross product

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%5Coverrightarrow%7Bb%7D%20%5Ctimes%20%5Coverrightarrow%7Bc%7D%20%3D%20%5Cbegin%7Bvmatrix%7D%20%5Chat%7Bi%7D%20%26%20%5Chat%7Bj%7D%20%26%20%5Chat%7Bk%7D%20%3Cbr%3E%20-1%20%26%201%20%26%202%20%3Cbr%3E%202%20%26%201%20%26%204%20%5Cend%7Bvmatrix%7D

Expanding this determinant:

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%3D%20%5Chat%7Bi%7D%20(4%20-%202)%20-%20%5Chat%7Bj%7D%20(-4%20-%204)%20%2B%20%5Chat%7Bk%7D%20(-1%20-%202)

%3D%20%5Chat%7Bi%7D%20(2)%20-%20%5Chat%7Bj%7D%20(-8)%20%2B%20%5Chat%7Bk%7D%20(-3)

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Step 3: Next, compute the dot product

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%3D%20(1)(2)%20%2B%20(2)(8)%20%2B%20(3)(-3)%20%3D%202%20%2B%2016%20-%209%20%3D%209

Step 4: Now calculate the volume:

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Thus, the volume of the tetrahedron is:

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Chapter Questions
1 MCQs