Solving Non-Collinear Points

In this section we have solving non-collinear points. In three dimensional geometry, collinear is one of the important topics to measure any of three points are collinear, planes, and so on. In below we will some problems in collinear and non-collinear. We also have some practice problems with answers without explanation. Let us see about solving non-collinear points.

Example problems for solving non-collinear points:

Example problem 1: Check that following points are collinear or non- collinear: P (–2, 3, 5), Q (1, 2, 3) and R (7, 0, –1).

Solution:

Given P (–2, 3, 5), Q (1, 2, 3) and R (7, 0, –1)

To check that given is collinear or non-collinear, we have to calculate the distance between the points.

If there is any relation between the points means collinear, otherwise non-collinear.

Now,

PQ = sqrt((1+ 2)^2+ (2 - 3)^2+ (3- 5)^2) =sqrt( 9 +1+ 4) = sqrt(14)

QR = sqrt((7 -1)^2+ (0 - 2)^2+ (-1-3)^2) = sqrt(36 + 4 +16) = sqrt(56) = 2 sqrt(14)

PR = sqrt((7 + 2)^2+ (0 - 3)^2+ (-1- 5)^2)= sqrt(81+ 9 + 36) = sqrt(126) = 3sqrt(14)

From the above we can see that, PQ + QR = PR.

Therefore, the P (–2, 3, 5), Q (1, 2, 3) and R (7, 0, –1) are collinear.

Answer: P (–2, 3, 5), Q (1, 2, 3) and R (7, 0, –1) are collinear.

Example problem 2: Are the following points A (3, 6, 9), B (10, 20, 30) and C (25, – 41, 5), the non-collinear?

Solution:

Given A (3, 6, 9), B (10, 20, 30) and C (25, – 41, 5)

To check that given is collinear or non-collinear, we have to calculate the distance between the points.

If there is any relation between the points means collinear, otherwise non-collinear.

 AB = sqrt((10 -3)^2 + (20- 6 )^2 + (30-9)^2) = sqrt(49 + 324 +441) = sqrt(814)

BC = sqrt((25 - 3)^2 + (-41- 20)^2 + (5 - 30)^2 )= sqrt(225 + 3721 + 625) = sqrt( 4571)

AC = sqrt((25 -3)^2 +(-41 - 6)^2 + (5 -9)^2) = sqrt(484 + 2209 + 16) = sqrt(2709) = 3sqrt(301)

Form the above analysis we can see that there is no relation between the three points. So the given are non-collinear.

Answer: The A (3, 6, 9), B (10, 20, 30) and C (25, – 41, 5) non-collinear.

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Practice problems for solving non-collinear points:

Practice problem 1: Show that the following points are non-collinear (2, –1, 3) (0, 1, -1) and (-2, 1, 3).

Practice problem 2: Check that the points (–2, 3, 5), (1, 2, 3) and (7, 0, –1) are collinear or non-collinear.

Solutions for solving non-collinear points:

Solution 1: Non-collinear

Solution 2: Collinear