Math Lines in three dimensions Intercepts

# Intercepts

The intersections of a line with the coordinate planes $E_{xy}$, $E_{xz}$, $E_{yz}$ are called intercepts.

!

### Remember

A line can have 1, 2, 3 or an infinite amount of intercepts.
i

### Method

1. Set corresponding coordinate equal to zero and determine $r$
2. Insert $r$ in the linear equation to get the intercepts
i

### Hint

The planes always have the coordinate zero, which does not appear in the name.
• $E_{xy}: z=0$
• $E_{xz}: y=0$
• $E_{yz}: x=0$

### Example

Determine the intercept of the line $g$ with the xy-plane.

$\text{g: } \vec{x} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + r \cdot \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix}$

1. #### Determine $r$

Since it is the plane $E_{xy}$, we set z equal to 0.

$\begin{pmatrix} x \\ y \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + r \cdot \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix}$

The row with just one variable (the third) has to be solved for $r$.

$0=3+6r\quad|-3$
$-3=6r\quad|:6$
$r=-0.5$

2. #### Determine intercept

$\text{g: } \vec{x} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + r \cdot \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix}$

The calculated $r=-0.5$ is used in the linear equation.

$\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + (-0.5) \cdot \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix}$ $=\begin{pmatrix} -1 \\ -0.5 \\ 0 \end{pmatrix}$

The intercept with the xy-plane is at $S_{xy}(-1|-0.5|0)$