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28 November, 12:24

Find the volume of the tetrahedron having the given vertices. (3, - 1, 1), (5, - 2, 4), (1, 1, 1), (0, 0, 1)

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  1. 28 November, 13:20
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    Given four vertices of a tetrahedron, we need to find its volume.

    A (3,-1,1), B (5,-2,4), C (1,1,1), D (0,0,1)

    The simplest method is to use vectors.

    Here, we calculate the volume of a parallelepiped defined by vectors AB, AC, AD. The volume of a tetrahedron is one-sixth of the volume of the parallelepiped, V, given by AB. (ACxAD).

    Here are the steps:

    1. Choose a vertex, say A. Calculate vectors AB, AC and AD

    AB = (5-2,4) - (3,-1,1) =

    AC = (1,1,1) - (3,-1,1) =

    AD = (0,0,1) - (3,-1,1) =

    2. Calculate vector cross product ACxAD = x

    i j k

    -2 - 2 0

    -3, 1, 0

    =

    =

    3. Calculate volume of parallelepiped, equal to the dot product AB. (ACxAD), ignore sign, which is arbitrary.

    V = |.| = |12| = 12

    4. Calculate volume of tetrahedron, which is one-sixth of volume of parallelepiped

    V (tetrahedron) = V/6 = 12/6 = 2
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