the point AB and C lie in the same plane relative to the ori

the point A,B and C lie in the same plane relative to the origin O, and have position vectors. a=3i-j+4k,b=-i+2j,c=5i-3j+7k respectively.(a) Find AB times AC.(b) Find the equation of the plane in the form (r times n =p) If the point D has position vector 5i+2j+3k. (c) Calculate the volume of the tetrahedron ABCD.

Solution

Given that A = 3i-j+4k, B = -i+2j, C = 5i-3j+7k , D = 5i+2j+3k

a)

AB = b-a = -i+2j - (3i-j+4k) = -4i +3j -4k

AC = c-a = 5i-3j+7k - ( 3i-j+4k) = 2i - 2j + 3k

Hence, we can say that

AB is not a multiple vector of AC.

c) volume of the tetrahedron ABCD

= (1/6) [ AB AC AD]

AB = -2i +3j -4k

AC = 2i - 2j + 3k

AD = D - A =  5i+2j+3k -  (3i-j+4k) = 2i+ 3j -k

Hence,

volume of the tetrahedron ABCD = (1/6) [ AB AC AD]

= (1/6) det of -2 3 -4

2 -2 3

2 3 -1

= (1/6) [ -2(2-9) -3(-2-6) -4(6+4) ]

= (1/6) [14 +24 -40]

= 1/3

Therefore,

volume of the tetrahedron ABCD = 1/3