Question

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

Short answer

Therefore, the parametric equations are $x=6-5t,y=7-10t,t\in [0,1]$

Step-by-step solution

Step: 1 Given information

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

Calculation

The goal is to determine the parametric equations for the line segment that connects the two points.

The formula for the line segment joining the pair of points to is as follows,

$x=a+(c-a)t,y=b+(d-b)t,t\in [0,1].$

Substitute the given values in the equation of the line segment to discover the parametric equations.

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

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

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

$$\begin{gathered} x=6+(-5)t \\ x=6-5t \end{gathered}$$

Step:3 Further calculation

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

Enter in the blanks in the equation. $y=b+(d-b)t,$we get $y=7+(-3-7)t$since $a=6,b=7,c=1,d=-3$

Enter in the blanks in the equation.

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

The parametric equations for the line segment connecting the two points are thus $(6,7),(1,-3)$are $x=6-5t,y=7-10t,t\in [0,1].$

Thus, the parametric equations are $x=6-5t,y=7-10t,t\in [0,1]$