Find the value of `lambda` if the volume of a tetrahedron whose vertices are with position vectors ` hat i-6 hat j+10 hat k ,- hat i-3 hat j+7 hat k ,5 hat i- hat j+lambda hat k and 7 hat i-4 hat j+7 hat k` is 11 cubic unit.

Answer

7

Explanation

Let the vertices be, A ,B , C , D and O be the origin.
`vecOA=hati -6hatj+10hatk,vecOB=hati-3hatj +7hatk`,
`vecOC= -5hati-hatj+lambdahatk,vecOD=7hati -4hatj+7hatk`
`vecAB=vecOB-vecOA= -2hati+3hatj-3hatk`
`vecAC=vecOC-vecOA= -4hati + 5hatj + (lambda-10)hatk`
`vecAC=vecOC -vecOA=4hati+5hatj+(lamda-10)hatk`
`vecAD=vecOD-vecOA = 6hati +2hatj-3hatk`
volume of tetrahedron
`1/6[vecAB vecAC vecAD]=1/6|{:(-2,3,-3),(4,5,lamda-10),(6,2,-3):}|`
`1/6 {-2(-15-2lambda+20)-3(-12-6lambda+60)-3(8-30)}`
`1/6 {4lambda- 10 -144 + 18 lambda+66}`
`= 1/6 (22lambda - 88) =11`
`or 2lambda -8 =6`
`or 2lambda -8 =6`
`or lambda=7`