three dimensional coordinate systems

[maeveoneill]

on a three dimensional coordinate system, how would you go about determining whether or not a group of points lie on a straight line.

eg. A (x1,y1,z1), B (x2,y2,z2), C (x3,y3,z3)

thanks

 

[stapel]

One method might be to find the distances between the pairs of points. If the sum of two of the distances equals the third, then the lines must be collinear.

Eliz.

 

[maeveoneill]

Determine whether the points lie on a straight line.
(a) A(2,4,2) B(3,7,-2) C(1,3,3)
i got
|AB| = 5.099
|AC|= 1.73
|BC|= 6.708

the answer for the back of this says they are not on a straight line

(b) D(0,-5,5) E(1,-2,4) F (3,4,2)
i got
|AB| = 3.316
|AC| = 9.949
|BC| = 6.48

the answer for this one says they are on a straight line

so i still dont get how this works.. becuase when calculating two sums for each.. they are both off about the same.. so why is one on a sraight line and not the other?!

 

[pka]

on a three dimensional coordinate system, how would you go about determining whether or not a group of points lie on a straight line. eg. A (x1,y1,z1), B (x2,y2,z2), C (x3,y3,z3)

Almost any question about lines involves slopes.
Is it true that:
\(\displaystyle \frac{{y_2 - y_1 }}{{x_2 - x_1 }} = \frac{{y_3 - y_1 }}{{x_3 - x_1 }}\)

Same slope means "on the same line'.

 

[maeveoneill]

okay i would understand doing that if they were on a two dimensional plane.. butt shouldnt there be something to incorporate teh z coordinate because its a three dimensional plane?

 

[pka]

I am sorry to say that I really did miss-read your question.
However, the answer is basically the same.

 

[stapel]

(a) A(2,4,2) B(3,7,-2) C(1,3,3)
i got
|AB| = 5.099
|AC|= 1.73
|BC|= 6.708

Is round-off error perhaps causing some problems...? :shock:

Try doing the calculations exactly, instead of in your calculator:

. . . . .|AB| = sqrt[(2 - 3)[sup:227j0n0n]2[/sup:227j0n0n] + (4 - 7)[sup:227j0n0n]2[/sup:227j0n0n] + (2 - (-2))[sup:227j0n0n]2[/sup:227j0n0n]] = sqrt[1 + 9 + 16] = sqrt[26]

. . . . .|AC| = sqrt[(2 - 1)[sup:227j0n0n]2[/sup:227j0n0n] + (4 - 3)[sup:227j0n0n]2[/sup:227j0n0n] + (2 - 3)[sup:227j0n0n]2[/sup:227j0n0n]] = sqrt[1 + 1 + 1] = sqrt[3]

. . . . .|BC| = sqrt[(3 - 1)[sup:227j0n0n]2[/sup:227j0n0n] + (7 - 3)[sup:227j0n0n]2[/sup:227j0n0n] + (-2 - 3)[sup:227j0n0n]2[/sup:227j0n0n]] = sqrt[4 + 16 + 25] = sqrt[45]

It's hard to "see" anything from this. For sqrt[26] + sqrt[3] to equal sqrt[45], their squares would have to be equal as well. But:

. . . . .(sqrt[26] + sqrt[3])[sup:227j0n0n]2[/sup:227j0n0n] = 26 + 2sqrt[78] + 9 = 35 + 2sqrt[78]

. . . . .(sqrt[45])[sup:227j0n0n]2[/sup:227j0n0n] = 45

Since sqrt[78] does not equal 5, then these cannot be equal, and the points cannot be collinear.

On the other hand:

(b) D(0,-5,5) E(1,-2,4) F (3,4,2)

. . . . .|DE| = sqrt[(0 - 1)[sup:227j0n0n]2[/sup:227j0n0n] + (-5 - (-2))[sup:227j0n0n]2[/sup:227j0n0n] + (5 - 4)[sup:227j0n0n]2[/sup:227j0n0n]] = sqrt[1 + 9 + 1] = sqrt[11]

. . . . .|DF| = sqrt[(0 - 3)[sup:227j0n0n]2[/sup:227j0n0n] + (-5 - 4)[sup:227j0n0n]2[/sup:227j0n0n] + (5 - 2)[sup:227j0n0n]2[/sup:227j0n0n]] = sqrt[9 + 81 + 9] = sqrt[99] = 3 sqrt[11]

. . . . .|EF| = sqrt[(1 - 3)[sup:227j0n0n]2[/sup:227j0n0n] + (-2 - 4)[sup:227j0n0n]2[/sup:227j0n0n] + (4 - 2)[sup:227j0n0n]2[/sup:227j0n0n]] = sqrt[4 + 36 + 4] = sqrt[44] = 2 sqrt[11]

Since 1 sqrt[11] + 2 sqrt[11] equals 3 sqrt[11], these points are indeed collinear.

Round-off error can be a killer! :wink:

Eliz.