Question

Calcular o volume do tetraedro abcd, sendo dados: a (1, 0, 0), b (0, 1, 0), c (0, 0, 1) e d (4, 2, 7).

Answer 1

ab = b - a => (0,1,0) - (1,0,0) = (-1,1,0)

ac = c - a => (0,0,1) - (1,0,0) = (-1,0,1)

ad = d - a => (4,2,7) - (1,0,0) = (3,2,7)

 

Produto misto ac x ad:

|i  j  k : i  j|

|-1 0 1 : -1 0| = [(0x7)-(1x2)]i + [(1x3)-(-1x7)]j + [(-1x2)-(0x3)]k

|3  2  7 : 3 2| = (0-2)i + [3-(-7)]k + (-2-0)k => -2i + 10j - 2k

                     = (-2, 10, -2)

 

V= 1/6 .ab .(ac x cd)

V= 1/6. (-1,1,0).(-2,10,-2) 

V= 1/6. (2 + 10 - 0)

V= 1/6 . 12

V= 12/6 => V= 2 u. v.