Let P = P(x, y, z)

Case 1:

P lies internally (between A and B)

if A(x1,y1,z1) and B(x2,y2,z2) and P(x,y,z) lies between them so that AP : PB = m1 : m2 the coordinates of P can by calculated using the following formulas:

x = (m1*x2 + m2*x1)/(m1 + m2)

y = (m1*y2 + m2*y1)/(m1 + m2)

z = (m1*z2 + m2*z1)/(m1 + m2)

in our case we have x1=0, y1=-1, z1=2, x2=3, y2=1, z2=-2, m1=3 and m2=2

x = (3*3 + 2*0)/(3 + 2)

y = (3*1 + 2*(-1))/(3 + 2)

z = (3*(-2) + 2*2)/(3 + 2)

x = 9/5 = 1 4/5 = 1.8

y = 1/5 = 0.2

z = -2/5 = - 0.4

the coordinates of P are P(1 4/5, 1/5, -0.4)

Case 2:

P lies externally (B is between A and P)

if A(x1,y1,z1) and B(x2,y2,z2) and P(x,y,z) lies on the line AB externally so that AP : PB = m1 : m2 the coordinates of P can by calculated using the following formulas:

x = (m1*x2 - m2*x1)/(m1 - m2)

y = (m1*y2 - m2*y1)/(m1 - m2)

z = (m1*z2 - m2*z1)/(m1 - m2)

in our case we have x1=0, y1=-1, z1=2, x2=3, y2=1, z2=-2, m1=3 and m2=2

x = (3*3 - 2*0)/(3 - 2)

y = (3*1 - 2*(-1))/(3 - 2)

z = (3*(-2) - 2*2)/(3 - 2)

x = 9

y = 5

z = -10

the coordinates of P are P(9, 5, -10)