Given 3 sets of ordered triples (for a 3D graph), how do you determine if the points make a straight line?

Question

Just started Calculus III, and I'm not exactly sure how to do this problem.

The exact question is:

Determine whether the points lie on a straight line

P(2, 4, 2), Q(3, 7, -2), R(1, 3, 3)

Any help is appreciated

Answer 1

In 3D, a line has the equation

(x-x1)/ a = (y-y1) /b = (z-z1)/c

a,b and c are constants to be determined and

(x1,y1,z1) are a point on the line.

Now two points determines a line, so you should be able to substitute 2 points in the above (one will be x1,y1,z1), the other (x2,y2,z2).  With those points you should be able to calculate a, b and c.  Now you can see if the third point with one of the others leads to the same a, b and c.  

Answer 2

This is a possible approach:

1)  Make a sketch of the 3D graph as well as you can.

2)  Looking at the xy, xz, and yz planes separately, determine whether the points form a straight line in each plane (that is, calculate the slope of PQ and QR in the xy, xz, and yz planes).

3)  If the points form a straight line in all the planes then it would be safe to say that they do indeed lie on a straight line.  

4)  Alternatively, you could plug the points into a graphing calculator or graph program and run a line-fitting program.  If your R^2 value is 1 then you have a straight line (though I doubt this approach would earn you even partial credit, you could use it to check your hand calculation).  

Hope this helps!