Example 1. Find the ratio in which the line joining the points (2, 4, 5), (3, 5, 4) is divided by the XY-plane.

Sol. Let the XY-plane divide the join of P(2, 4, 5) and Q(3, 5, 4) in the ratio k : 1. Then the coordinates of the point of division are .

Since this point lies on the XY-plane

? its z coordinate is zero.

?

Hence the line joining (2, 4, 5) and (3, 5, 4) is divided by the XY-plane internally in the ratio 5 : 4.

Example 2. Given that P(3, 2, 4), Q(5, 4, 6) and R(9, 8, 10) are collinear, find the ratio in which Q divides PR.

Sol. Let Q(5, 4, 6) divide PR in the ratio k : 1

Then the coordinates of Q are .

But the coordinates of Q are given to be (5, 4, 6).

? Equating the x-coordinates, we have

Hence Q divides PR internally in the ratio 1/2 : 1 or 1 : 2.

Note. The same value of k is obtained when we equate the y or z coordinates.

Example 3. A(3, 2, 0), B(5, 3, 2), C( 9, 6, 3) are three points forming a triangle. AD, the bisector of angle BAC, meets BC in D. Find the coordinates of the point D.

Sol. Since AD is the internal bisector...