If ` vec a , vec b , vec ca n d vec d` are four vectors in three-dimensional space with the same initial point and such that `3 vec a+2 vec b+ vec c-2 vec d=0` , show that terminals `A ,B ,Ca n d D` of these vectors are coplanar. Find the point at which `A Ca n dB D` meet. Find the ratio in which `P` divides `A Ca n dB Ddot`

Explanation

Since `3veca-2vecb+vecc-2vecd=vec0`
`" "3veca+vecc= 2vecb+ 2vecd`
`rArr" "(3veca+vecc)/(4)= (2vecb+ 2vecd)/(4) or (3vec a+2vecd)/(3+1)= (vecb+vecd)/(2)`
Therefore, P.V. of the point dividing AC in the ratio `1 : 3` is the same as the P.V. of midpoint of BD.
So AC and BD intersect at P, whose P.V. is `(3veca+vecc)/(4) or (vecb+vecd)/(2)`. Point P divides AC in the ratio `3 : 1` and BD in the ratio `1 : 1`.