Ecuaciones de la recta en el plano

Parametric equation

Ecuación vectorial
The vector equation:
$/fs1/vec{OX}=/vec{OP}+/lambda/vec{u}$
is the key to finding the parametric equation of a straight line. If we solve this equation for the vector OX:

$/fs1/vec{OX}=/vec{OP}+/lambda/vec{u}/;/Rightarrow/;(x,y)=(p_1,p_2)+/lambda(u_1,u_2)$
$/fs1/Rightarrow/;(x,y)=(p_1/;+/;/lambda/;u_1/;,/;p_2/;+/;/lambda/;u_2)$

and write the above vector equation as two separate scalar equations we have:

$/left/{{/;/;x/;=/;p_1/;+/;/lambda/;u_1/;/atop/;y/;=/;p_2/;+/;/lambda/;u_2}$ (parametric equation of a straight line)

Parametric equation of a straight line

$/left/{{/;/;x/;=/;p_1/;+/;/lambda/;u_1/;/atop/;y/;=/;p_2/;+/;/lambda/;u_2}$

Given the vector equation: $/fs1(x,y)=(2,3)+/lambda(-2,1)$ of a straight line, find the parametric equation:

Firstly, $/fs1(x,y)=(2,3)+/lambda(-2,1)/;/Rightarrow/;(x,y)=(2-2/lambda,3+/lambda)$

Then the parametric equation is:

$/left/{{/;/;x/;=/;2/;-2/lambda/;/atop/;y/;=/;3/;+/;/lambda}$

Find the parametric equation of the line that passes through the points
P(1,-1) and Q(0,-3).

The vector PQ is: $/fs1/vec{PQ}=(0-1,-3-(-1))=(-1,-2)$ .
Then:

$/left/{{x/;=/;1/;-/lambda/;/atop/;y/;=/;-1/;-2/lambda}$

Find the parametric equation of the line that passes through the points $/fs1P(-3,9)$ and $/fs1Q(4,-4)$

x= +

y= +