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.

Answer 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).