Vector Lines

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

$$\lambda = \frac{x - a_1}{b_1} = \frac{y - a_2}{b_2} = \frac{z - a_3}{b_3}$$

example


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