SOLUTION: Determine whether the three points P(-6, -9, 4), Q(-8, -13, -2) and R(-10, -17, -10) are collinear by computing the distances between pairs of points. (a) Distance from P to Q

Question 833927: Determine whether the three points P(-6, -9, 4), Q(-8, -13, -2) and R(-10, -17, -10) are collinear by computing the distances between pairs of points.
(a) Distance from P to Q:
(b) Distance from Q to R:
(c) Distance from P to R:
(d) Are the three points colinear (y/n)?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
This is one ugly problem.
Calculating distances is a cumbersome way to find if the points are colinear.
The answer for part (d) could be more easily found in different ways.
I can see they are not colinear by simple mental math, but to answer parts (a), (b), and (c) we need to calculate those distances.
Maybe the problem was designed to make students practice concentration, arithmetic and patience,
or maybe students were taught to use a fancy calculator/computer programs to calculate distances without having to think.
I don't like it. I like thinking, understanding, and solving problems by the shortest, more direct route.
However, no one else seemed to want to solve this problem, so I might as well do it.

If P, Q and R are colinear, two of the distance will add up to the third distance.
If not, the points will be forming a triangle and the sum of the two shortest distances will be greater than the longer distance.

CALCULATING DISTANCES:
The distance between two points is
the square root of
the sum of the squares of
the differences in the points coordinates.

NOTE: That is an ugly mouthful that correspond to the uglier formula below,
but the concept is simple and neat;
just apply the Pythagorean theorem once if given two variables,
twice if given 3 variables.
If this NOTE makes sense to you, enjoy it. If it does not make sense, ignore it.

The distance between points and is

Similarly,

and

I do not see any benefit from working with those pesky square roots, so let's add the approximate values. They are correct to within ,
so assuming the sum of the shortest distance added to exactly the longest distance,
we could not be off by more than .
is the sum,
and it differs from by ,
so the points are not colinear.
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