How to show whether three points in $\mathbb{R^4}$ lie on a straight line?

Question

If you are given the coordinates of three points in $\mathbb{R^4}$. (Call these three points: A, B and C). How do we know whether these three points lie on a straight line or not?

One way I can think about it is to find the coordinates of the two vectors: $AB$ and $AC$ and see whether they are a constant multiple of each other. If I can show that they are constant multiple of each other, then lie on the same straight line; Otherwise, they don't lie on the same straight line.

Is there any other way to do this problem? Thanks!

Answer 1

I'd parametrize the straight line passing between two of them and see if it passes through the third:

Consider $\gamma(t) = tA + (1-t)B$. The you want to see if there exists a $t_0\in\mathbb{R}$ such that $\gamma(t_0) = C$.

Answer 2

"The points a, b and c are collinear if and only if d(x,a) = d(c,a) and d(x,b) = d(c,b) implies x=c." - Wikipedia

This seems like the easiest way to me. Were you given a definition of collinear?

Answer 3

One may parametrize a line between two given points as follows:

$$\alpha(t) = P_0 + (P_1 - P_0)t$$

Note that $\alpha(0) = P_0$ and $\alpha(1) = P_1$.

Now suppose you are given a third point $P_2$. To check if $P_2$ lies on the line, simply check whether there is a $t$ such that $\alpha(t) = P_2$.