Question

Question 4 Using section formula, show that the points A (2, –3, 4), B (–1, 2, 1) and are collinear.

Class X1 - Maths -Introduction to Three Dimensional Geometry Page 277

Answer 1

Let C divides AB in the ratio K : 1
A----------------------C--------------B
∵ x - co-ordinate of C is zero ,
∴ we have to use section formula for x - co-ordinate
x = (mx₂ + nx₁)/(m + n )
Here, m : n = k : 1
x₂ = -1 and x₁ = 2

Now,
0 = (k × -1 + 1 × 2)/( k + 1)
0 = -k + 2
K = 2

use section formula for y - co-ordinate
y = (my₂ + ny₁)/(m + n)
Here, m : n = k : 1
y₂ = 2 and y₁ = -3
Now,
1/3 = ( k × 2 + 1 × -3)/( k + 1)
k + 1 = 2k - 3
k = 2

Similarly you will get in z - co-ordinate k = 2


Hence, C divides AB internally in the ratio 2 : 1 .hence, Point A, B and C are collinear .