Prove that 3 points are not on the same line

Question

Given $P_1=(1, 1, 1)$, $P_2=(2, -1, 2)$ and $P_3=(3,0,1)$, I need to prove that these three points are not on the same line.

What I tried - I showed that $\vec{P_1P_2}$, $\vec{P_1P_3}$ and $\vec{P_2P_3}$ are different (meaning they have different directions?). Does that conclude the proof?

Thanks.

Answer 1

The points $P_1$, $P_2$ and $P_3$ are not on the same line iff $\vec{P_1P_2}$ and $\vec{P_1P_3}$ are not colinear (ie. there exists $\lambda \in \mathbb{R}$ such that $\vec{P_1P_2} = \lambda \vec{P_1P_3}$).

For example, $A=(0,0,0)$, $B=(1,1,1)$ and $C=(2,2,2)$ are on the same line, but $\vec{AB}$ and $\vec{AC}$ are different; however, $\vec{AC}=2 \vec{AB}$.

Answer 2

Let slope of $\vec{P_1P_2} = m_1$ and slope of $\vec{P_2P_3} = m_2$, you see that $m_1 \neq m_2$.

Aliter:

$|P_1P_3|=|P_1P_2|+|P_2P_3| \implies $ The points are in straight line,else they are not.

Here $|xy|$ denotes the distance from point $x$ and $y$.