Question

In Problems 1 to 5, all lines are in the $\left(x,y\right)$ plane.

3. Write, in parametric form [as in Problem 1], the equation of the straight line that joins $\left(1,-2\right)$and $\left(3,0\right)\mathbf{.}$

Short answer

The parametric equation of line is $\left(3i\right)+(i+j)t$.

Step-by-step solution

Concept of the straight line in slope form.

Write the expression of slope.

$\frac{\mathbf{y}\mathbf{-}{\mathbf{y}}_{\mathbf{o}}}{\mathbf{x}\mathbf{-}{\mathbf{x}}_{\mathbf{o}}}\mathbf{=}\mathit{m}$ ………… (1)

Here,$\mathit{m}$ is the slope and${\mathit{x}}_{\mathbf{0}}$ and${\mathit{y}}_{\mathbf{0}}$are the coordinates.

Substitute the values in slop equation 1 and 2

First point is$(1,-2)$ and the second point is$(3,0)$.

Substitute -2 for $y_0$,1 for $x_0$,0 for $y_1\sqrt{2}$ and 3 for $x_1$, in equation (1).

$$\begin{gathered} \frac{0-(-2)}{3-1}=m \\ m=1 \end{gathered}$$

Substitute 0 for $y_o,$ 3 for $x_o,$ and 1 for $m$ in equation (1).

$$\begin{gathered} \frac{y-0}{x-3}=1 \\ \frac{y}{1}=\frac{x-3}{1} \end{gathered}$$

Thus, $a=1$ and $b=1$ in parametric form.

Substitute the values in parametric form

The parametric form is,$x=3+t$.

And,$y=t$.

Thus, the parametric equation can be written as follows:

$r=(3,0)+(1,1)t$

Therefore, the parametric equation of the line is $\left(3i\right)+(i+j)t$.