(i) Let P(x, y, z) be any point which divides the line segment joining the points A(–2, 3, 5) and B(1, –4, 6) in the ratio 2 : 3 internally.
Then x =

2×1+3×(−2)

2+3

=

2−6

5

= −

4

5

, y =

2×(−4)+3×3

2+3

=

−8+9

5

=

1

5

z =

2×6+3×5

2+3

=

12+15

5

=

27

5

∴ Coordinates of P are (−

4

5

,

1

5

,

27

5

)
(ii) Let P(x, y, z) be any point which divides the line segment joining the points A(–2, 3, 5) and B(1, –4, 6) in the ratio 2 : 3 externally.
Then
x =

2×1−3×(−2)

2−3

=

2+6

−1

= - 8 , y =

2×(−4)−3×3

2−3

=

−8−9

−1

= 17
z =

2×6−3×5

2−3

=

12−15

−1

= 3
∴ Coordinates of P are (–8, 17, 3)