Math Lines in three dimensions Point on the line

# Point on the line?

This is to check whether a point lies on a given line.

i

### Method

1. Insert the position vector of the point for $\vec{x}$ into the linear equation
2. Set up equation system (one equation per row)
3. Check if $r$ is the same for each row

### Example

Is the point $A(-3|14|10)$ located on the line $g$?.

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

1. #### Insert $A$ in $g$

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

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

2. #### Set up equation system

Now we set up the 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

If there is no contradiction and $r$ is equal in all equations, then the point is on the line.

I, II, III: $r=2$

=> Point $A$ is located on the line.