The base of a triangular pyramid is a triangle with corners at #(3 ,4 )#, #(2 ,9 )#, and #(5 ,7 )#. If the pyramid has a height of #7 #, what is the pyramid's volume?

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Answer 1 · 2016-08-06T19:51:34

#91/6#

Given

#p_1 = {3,4,0}#
#p_2 = {2,9,0}#
#p_3 = {5,7,0}#

and deffining

#vec u = p_1-p_2# and
#vec v = p_3-p_2#

The area spanned by two vectors #vec u, vec v# with the same origin, has a vectorial formula.

#A = 1/2 norm(vec u xx vec v) = 13/2#

The pyramid volume is given by

#V = 1/3 h A = 1/3 cdot 7 cdot 13/2 = 91/6#