vector equation of a line
In 3-dimensional space a vector can be defined by the position of one point on the line and the direction vector from that point which is usually calculated knowing the position of a second point on the line.
$A(a_1,a_2,a_3)$
and $B(b_1, b_2, b_3)$
. The position vector of A is $\vec{OA}$
and the direction vector is for example $\vec{AB}$
.
any point $P(x,y,z)$
on the line g is then defined by
$$g : \vec{x} = \vec{a} + \lambda \cdot (\vec{b} - \vec{a})$$
$${g : \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} + \lambda \cdot \begin{pmatrix} b_1 - a_1 \\ b_2 - a_2 \\ b_3 - a_ 3 \end{pmatrix}}$$
where $\lambda$
is the parameter that determines the distance of any point X on the line from point A as a factor of the magnitude (length) of the vector $\vec{AB}$
.
parametric equation
$x = a_1 + \lambda \cdot b_1$
$y = a_2 + \lambda \cdot b_2$
$z = a_3 + \lambda \cdot b_3$
Cartesian equation
Solving each expression for $\lambda$
, we get
$$\lambda = \frac{x - a_1}{b_1} = \frac{y - a_2}{b_2} = \frac{z - a_3}{b_3}$$