Line

Topics: Line - 3D Space

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Let $p=(x_1,x_2,x_3)$ be a point in ${\mathbb{R}}^3$ and $V=(v_1,v_2,v_3)$ a non-zero vector in ${\mathbb{R}}^3$.

Vectorial Representation §

In its vectorial representation, the line that goes through the point $P$ in the direction of the vector $V$ is the set:

$l=\{P+tV\mid t\in \mathbb{R}\}$

This set can also be written as follows:

$$l=\{(x_1,x_2,x_3)+t(v_1,v_2,v_3)\mid t\in \mathbb{R}\}$$

$$l=\{(x_1+t{v}_1,x_2+t{v}_2,x_3+t{v}_3)\mid t\in \mathbb{R}\}$$

Parametric Representation §

In its parametric representation, the line that goes through the point $P$ in the direction of the vector $V$ is:

$$l=\{\begin{array}{l}x=x_1+t{v}_1\\ y=x_2+t{v}_2\\ z=x_3+t{v}_3\end{array}$$

Symmetric Representation §

In its symmetric representation, the line that goes through the point $P$ in the direction of the vector $V$ is:

$$\frac{x-x_1}{v_1}=\frac{y-x_2}{v_2}=\frac{z-x_3}{v_3}$$

Beware!

This representation only works if $v_1,v_2,v_3\ne 0$.

If there is no $t$

If you have a line that doesn’t have $t$ in one of its arguments, then the symmetric representation changes.

For example, the following line:

$$l=\{\begin{array}{l}x=2\\ y=-9-9t\\ z=3-3t\end{array}$$

…as the symmetric representation:

$$\frac{y+9}{-9}=\frac{z-3}{-3};x=2$$

And the line:

$$l=\{\begin{array}{l}x=5\\ y=9-9t\\ z=-10\end{array}$$

…has the symmetric representation:

$$t=\frac{y-9}{-9};x=5;z=-10$$

Line Through the Origin §

When $P=(0,0,0)$, then we call such a line a line that goes through the origin. This line in its vectorial representation is:

$$l=\{tV\mid t\in \mathbb{R}\}$$

$$l=\{(t{v}_1,t{v}_2,t{v}_3\mid t\in \mathbb{R})\}$$

Line Through Two Points §

The line that passes through two points $P=(x_1,x_2,x_3)$ and $R=(y_1,y_2,y_3)$ is, in its vectorial representation:

$$l=\{P+t(R-P)\mid t\in \mathbb{R}\}$$