Find the volume of a tetrahedron with vertices:O(0,0,0), A(1,2,3), B(-2,1,5), C(3,7,1) by using triple integralFirst find the the equations of the planes.

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periasemdy · Accepted Answer

Step 1We can transform the points A(1,2,3), B(-2,1,5), C(3,7,1) to A′(1,0,0), B′(0,1,0), C′(0,0,1). So that the volume of the new tetrahedron is easy to compute. The transformation matrix from A′,B′,C′ to A,B,C is:  T  =      (                            1                          −          2                          3                                      2                          1                          7                                      3                          5                          1                      )    .Step 2Because the linearity this is also the Jacobian matrix, so      V    o    l    u    m    e    =      ∭          O      A      B      C        1    d  x  d  y  d  z  =      ∭          O              A        ′                    B        ′                    C        ′              1              |        det    (    T    )          |          d      x    ′    d      y    ′    d      z    ′    =      17    2    .

Find the volume of a tetrahedron with vertices: O(0,0,0) , A(1,2,3), B(-2,1,5), C(3,7,1) by using triple integral.

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