Question

In Problems 1 to 5, all lines are in the $\mathbf{(}\mathit{x}\mathbf{,}\mathit{y}\mathbf{)}$ plane.

1. Write the equation of the straight line through$\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{}\mathbf{-}\mathbf{}\mathbf{3}\mathbf{)}$ with slope $\mathbf{3}\mathbf{/}\mathbf{4}$, in the parametric form$\mathit{r}\mathbf{=}{\mathit{r}}_{\mathbf{0}}\mathbf{+}\mathit{A}\mathit{t}$.

Short answer

The parametric equation of the line is $(2,-3)+(4,3)t$.

Step-by-step solution

Concept of the straight line in slope form.

Write the expression of the straight line in slope form.

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

Here, is the slope and and are the coordinates.

Find the equation of the Straight line

The given point is $(2,-3)$ and the slope of line is $\frac{3}{4}$.

Substitute -3 for$y_o,2$for$x_o$and$\frac{3}{4}$for$m$in equation (1).

$$\begin{gathered} \frac{y-(-3)}{x-\left(2\right)}=\frac{3}{4} \\ \frac{y+3}{3}=\frac{x-2}{4} \end{gathered}$$

Thus, from the obtained equation, $a=4$ and $b=3$ in parametric form.

Substitute the value in parametric form

The parametric form is,$x=2+4t$.

And,$y=-3+3t$.

Thus, the parametric equation can be written as follows:

$r=(2,-3)+(4,3)t$

Therefore, the parametric equation of the line is $(2,-3)+(4,3)t$.