Question

Q. 2: Show that the points A (2, 3, – 4), B (1, – 2, 3) and C (3, 8, – 11)are collinear.​

Answer 1

Solution :-

We know the direction ratios of the line passing through two points P(x1, y1, z1) and Q(x2, y2, z2) are given by:

=> x2 – x1, y2 – y1, z2 – z1 or x1 – x2, y1 – y2, z1 – z2

Given points are A (2, 3, – 4), B (1, – 2, 3) and C (3, 8, – 11).

Direction ratios of the line joining A and B are

=> 1 – 2, – 2 – 3, 3 + 4 i.e., – 1, – 5, 7.

The direction ratios of the line joining B and C are

=> 3 –1, 8 + 2, – 11 – 3, i.e., 2, 10, – 14.

From the above, it is clear that direction ratios of AB and BC are proportional.

That means AB is parallel to BC. But point B is common to both AB and BC.

Hence, A, B, C are collinear points.

Answer 2

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We know that the direction ratios of the line passing through two points P(x1, y1, z1) and Q(x2, y2, z2) are given by:

x2 – x1, y2 – y1, z2 – z1 or x1 – x2, y1 – y2, z1 – z2

Given points are A (2, 3, – 4), B (1, – 2, 3) and C (3, 8, – 11).

Direction ratios of the line joining A and B are:

1 – 2, – 2 – 3, 3 + 4

i.e. – 1, – 5, 7.

The direction ratios of the line joining B and C are:

3 –1, 8 + 2, – 11 – 3

i.e., 2, 10, – 14.

From the above, it is clear that direction ratios of AB and BC are proportional.

That means AB is parallel to BC. But point B is common to both AB and BC.

Hence, A, B, C are collinear points

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