Prove by direction numbers, that the point(1, -1, 3), (2, -4, 5) and (

1.	Prove by direction numbers, that the point(1, -1, 3), (2, -4, 5) and (5, -13, 11) are in a straight Line.
Answer» Let the three given points are A(1, -1, 3), B(2, -4, 5) & C(5, -13, 11). The direction ratios of the line AB joining the points A and B : 2 - 1, -4 - (-1), 5 - 3 = 2 - 1, -4 + 1, 5 - 3, 2 ...(1) Similarly, the direction ratios of the line BC joining the points B and C: 5 - 2, -13 - (-4), 11 - 5 = 3, -9 ,6 ...(2) It is clear from (1) and (2) that direction ratios of AB and BC are proportional i.e. AB is parallel to BC. But the point B is common to both AB and BC. Therefore A, B and C are co-linear points.

Prove by direction numbers, that the point(1, -1, 3), (2, -4, 5) and (5, -13, 11) are in a straight Line.

Answer»

Let the three given points are A(1, -1, 3), B(2, -4, 5) & C(5, -13, 11). The direction ratios of the line AB joining the points A and B :

2 - 1, -4 - (-1), 5 - 3 = 2 - 1, -4 + 1, 5 - 3, 2 ...(1)

Similarly, the direction ratios of the line BC joining the points B and C:

5 - 2, -13 - (-4), 11 - 5 = 3, -9 ,6 ...(2)

It is clear from (1) and (2) that direction ratios of AB and BC are proportional i.e. AB is parallel to BC. But the point B is common to both AB and BC. Therefore A, B and C are co-linear points.

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