Express a point as an affine combination of another two points(3D collinearity)

Question

So, given the points A(1,2,2), B(2,4,2) and C(3,6,2) I have to show that they are collinear. If they are collinear then I must express one point as an affine combination of the other two points. I have searched everywhere and I can't find an explicit solve of such geometry problems, can you guys show me how I must proceed? Thank you.

score 1 · Accepted Answer · answered Oct 22 '14 at 10:47

1

We would like to find $x$ and $y$ such that

$$C = xA + yB$$

and

$$x + y = 1$$

To do this, we can simply pick the first coordinates of all three points. We then have the following equality:

$$1x + 2y = 3$$

Putting this together with $x + y = 1$, we have $x = -1$ and $y = 2$, and since you can verify that

$$C = -A + 2B$$

then $C$ is an affine combination of $A$ and $B$ and thus all three points are collinear.

(of course if you only care for collinearity, there are easier tests such as the cross product test).

answered Oct 22 '14 at 10:47

Irvan

2,208

Thanks, I now understand how this works, greatly appreciate it, I will vote up and accept the answer. – southpaw93 Oct 22 '14 at 10:58
quick question, how do I check that C=-A+2B ? – southpaw93 Oct 22 '14 at 12:06
Just compute $-A + 2B $ and see if it's equal to C? – Irvan Oct 22 '14 at 13:49
I understand, but what do I do in this case A(1,0,0),B(1,1,1),C(1,3,3) ? It won't work, and they are collinear. – southpaw93 Oct 23 '14 at 07:48
1

You have to pick a coordinate that's different. In this case use the y coordinates – Irvan Oct 23 '14 at 08:40

Express a point as an affine combination of another two points(3D collinearity)

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