Question

In Exercises 55–60 use the result of Exercise 54 to find parametric equations for the line segments connecting the given pairs of points in the direction indicated.

From$(1,-3)to(6,7)$

Short answer

The parametric equations are$x=1+5t,y=-3+10t,t\in [0,1]$

Step-by-step solution

Given information

The points are from$(1,-3)to(6,7)$

Calculation 

Consider the points $(1,-3)$to $(6,7)$.

The objective is to find the parametric equations for the line segment joining the pair of points.

The formula for the line segment joining the pair of points $(a,b)to(c,d)$is as follows, $x=a+(c-a)t,y=b+(d-b)t,t\in [0,1].$

Here, to find the parametric equations, substitute the given values in the equation of line segment.

Now take the points $(1,-3),(6,7)$.

Substituting the values in the equation $x=a+(c-a)t$we get,

$x=1+(6-1)t$

$x=1+5t$since $a=1,b=-3,c=6,d=7$

Now take the points $(1,-3),(6,7).$

Substitute the values in the equation $y=b+(d-b)t$we get

$$\begin{gathered} y=-3+(7-(-3\left)\right)t \\ y=-3+(7+3)t \\ y=-3+10t \end{gathered}$$

Thus, the parametric equations for the line segment joining the pair of points $(1,-3)(6,7)$are $x=1+5t,y=-3+10t,t\in [0,1].$

Therefore, the parametric equations are$x=1+5t,y=-3+10t,t\in [0,1]$.