Point on a line segment?

Checking whether a point is on a line segment is like finding out if a point is on a line but with one additional condition.

Example

Is the point $P(-3|14|10)$ on the line segment $\overline{AB}$ ?

$A(3|4|6)$ and $B(0|9|8)$

1. Set up $g_{AB}$

   Set up the linear equation from two points.

   $\text{g: } \vec{x} = \vec{OA} + r \cdot \vec{AB}$

   $\text{g: } \vec{x} = \begin{pmatrix} 3 \\ 4 \\ 6 \end{pmatrix} + r \cdot \begin{pmatrix} -3 \\ 5 \\ 2 \end{pmatrix}$

2. Point on the line

   The position vector (vector with the coordinates of the point) of $A$ must be inserted for $\vec{x}$ in $g$.

   $\begin{pmatrix} -3 \\ 14 \\ 10 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \\ 6 \end{pmatrix} + r \cdot \begin{pmatrix} -3 \\ 5 \\ 2 \end{pmatrix}$

   Now we set up an equation system and solve it. Every row is an equation.

   1. $-3=3-3r$
   2. $14=4+5r$
   3. $10=6+2r$
   
   
   1. $r=2$
   2. $r=2$
   3. $r=2$

3. Check

   $r=2$ is the case in all equations, so the point is on the line $g_{AB}$. However, the point is not on the line segment $\overline{AB}$, because $r>1$.

   => The point $P$ ist not located on the line segment.