Find Vector and Parametric Equation

Question

I'm having some trouble finding answers to these problems. When I try to find help online, all I find are $(x,y,z)$ problems and I'm simply looking for a PreCalculus $(x,y)$ problem solving technique:

The question is:

Find Vector and parametric equations of a line passing through $(3,1)$ and $(-4, -4)$

Answer 1

Let $\mathbf{P} = (3,1)$ and $\mathbf{Q} = (-4,-4)$. Then a vector equation for the line is $$ \mathbf{X}(t) = (1-t)\mathbf{P} + t \mathbf{Q} $$ You can easily check that $\mathbf{X}(0) = \mathbf{P}$ and $\mathbf{X}(1) = \mathbf{Q}$.

Alternatively, you could define a vector $\mathbf{V} = \mathbf{Q} - \mathbf{P} = (-7,-5)$, and then the vector equation could be written as $$ \mathbf{X}(t) = \mathbf{P} + t \mathbf{V} $$

Separating out the $x$ and $y$ coordinates, you get the parametric equations$$ x(t) = (1-t)(3) + t(-4) = 3 - 7t \\ y(t) = (1-t)(1) + t(-4) = 1 - 5t \\ $$

Actually, the two vector equations are "parametric" equations, too -- the terminology seems strange, to me, but I assume that this is what you want.

Answer 2

To find parametric equations $x(t)$ and $y(t)$ you need a vector and a point. Notice you already have $2$ points that you can use. Do you know remember how to find a vector?

$v = \langle x_2-x_1, \ y_2-y_1 \rangle$

Then use the form $x(t) = x_1 + t \ v_x$ and $y(t) = y_1 + t \ v_y$