If `P(x,y,z)` is a point on the line segment joining Q(2,2,4) and R(3,5,6) such that the projection of `vec(OP)` on the axes are `13/9, 19/5, 26/5` respectively, then P divides QR in the ratio:

Explanation

Since `vec(OP)` has projections `(13)/(5), (19)/(5) and (26)/(5)` on the coordinates axes, `vec(OP)=(13)/(5)hati+(19)/(5)hati+(26)/(5)hatk`. Suppose P divides the join of `Q(2, 2, 4) and R(3, 5, 6)` in the ratio `lamda:1`. Then the position vector of P is
`" "((3lamda+2)/(lamda+1))hati+((5lamda+2)/(lamda+1))hatj+((6lamda+4)/(lamda+1))hatk`
`therefore" "(13)/(5)hati+(19)/(5) hatj+(26)/(5) hatk`
`" "((3lamda+2)/(lamda+1))hati+((5lamda+2)/(lamda+1))hatj+((6lamda+4)/(lamda+1))hatk`
Thus, we have
`" "(3lamda+2)/(lamda+1)=(13)/(5), (5lamda+2)/(lamda+1)=(19)/(5) and (6lamda+4)/(lamda+1)=(26)/(5)`
`rArr" "2lamda=3 or lamda = 3//2`
Hence, P divides QR in the ratio 3 : 2.