Question

Find the volume of the parallelopiped, if the coterminus edges are given by the vectors `2hat"i" + 5hat"j" -4 hat"k", 5hat"i" +7hat"j"+5 hat "k" , 4hat"i" +5hat"j" - 2 hat"k"`.                               

Answer 1

 Let the given vectors 
` overline(a)=  2hat"i" + 5hat"j" -4 hat"k", overline(b) =  5hat"i" +7hat"j"+5 hat "k" ,overline(c) = 4hat"i" +5hat"j" - 2 hat"k"`.
represent the co-terminous edges of the parallelopied
Consider `overline("a") . (bar "b" xx bar"c") = abs[(2 ,5,-4),(5 , 7, 5),(4 , 5, -2)]`
= 2 (-14-25) -5 (-10-20) -4 (25 - 28)
= - 78 +150 + 12 = 84
∴Volume of the parallelopiped  = `abs(bar "a" .(bar"b" xx bar"c"))`
= 84 cubic units.