I need to find a parametric equation and a vector form for the lines in $\mathbb{R}^2$ with the equation
$$y = 3x – 1.$$
I know that the parametric and associated vector form is x=p+t d.
Could someone find a parametric equation and a vector form of this equation and could you explain how you found it?
Thank you.
Best Answer
The slope of a line is given by $m=\frac{\Delta y}{\Delta x}$. In this case, since the slope of the line is $3$, we have the equivalent direction vector $\vec{d}=(\Delta x,\Delta y)=(1,3)$. Since the given line has a $y$-intercept of $-1$, we know the line passes through the point with coordinates $(0,-1)$, and this gives us the final answer in its vector equation $$l:(x,y)=(0,-1)+t(1,3)$$ and in its parametric equations $$\left.l:\begin{cases} x=t\\ y=-1+3t \end{cases}.\right.$$