Proving a point lies between two others

Question

how can I show that the point $Q = (2, 5, -3)$ lies between $P(1, 6, -5)$ and $R(4, 3, 1)$? I have already proved that the three points are collinear, but I would like to show which is in the middle. Is there any way, other than drawing a picture, that can prove this fact?

Answer 1

$R$ is a candidate to be between $P$ and $R$.

To check it, using barycentric coordinates, you have to show the system of equations in $t$ $$tP+(1-t)R=Q\iff\begin{cases}t+4(1-t)=2\\6t+3(1-t)=5\\-5t+1-t=-3 \end{cases}$$ has a solution $t\in[0,1]$

Answer 2

We have that the line PR is given by

$$P+t(R-P)=(1, 6, -5)+t(3, -3, 6)$$

then check whether $\exists t$ such that

$$(2, 5, -3)=(1, 6, -5)+t(3, -3, 6)$$

and recall that

• $t=0$ gives $P$
• $t=1$ gives $R$
• $0<t<1$ the point lies between $P$ and $R$

Answer 3

Let's first find the line that goes through these three points. The line can be represented as $$(x,y,z)=(1,-1,2)\cdot t + (2,5,-3)$$

So, what values of $t$ produce each one of the points? $t=0$ obviously produces $Q$, $t=-1$ produces $P$ and $t=2$ produces $R$. Since $-1<0<2$, we are done.

Answer 4

Since you have already established the collinearity of your points, you can just compare the $x$ values.

You have $$1<2<4$$

Thus $Q$ is between $P$ and $R$