How to see the fact that the curve is the segment between $(x_1,...,x_n)$ and $(y_1,...,y_n)$

Question

It says that:

If $C = {x_1 + t(y_1- x_1),...,x_n + t(y_n- x_n);t ∈[0,1]}$ then C is the segment between $(x_1,...,x_n)$ and $( y_1,...,y_n)$

But how to see the fact? It seems to be not such intuitive. Could someone explain?

Thanks in advance!

Answer 1

If you rewrite as $$C(t) = [(1-t) x_1 + t y_1, \ldots , (1-t) x_n + t y+n], $$ then for $t = 0$, you can see that $C(0) = x$; and for $t = 1$, you can see that $C(1) = y$. And for $t = 0.5$, you can see that $C(0.5)$ is the average of $x$ and $y$. The same argument goes for any number: $C$ represents a weighted average between the two points.

Now it may not be obvious that the set of all weighted averages constituted a line, but in $n$ dimensions, what is your notion of what constitutes a line? Once you tell us what would convince you something is a line, we can perhaps connvince you that the set of all $C(t)$, as $t$ ranges over the reals, actually is one of these. (For this to be true, however, you need that $x \ne y$.)