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$(-3,5)to(1,4)$

Short answer

The required parametric equations are$x=-3+4t,y=5-t,t\in [0,1]$

Step-by-step solution

Given Information

The points from$(-3,5)to(1,4)$

Calculation

Consider the points $(-3,5)$to $(1,4).$

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 $(-3,5)$to $(1,4)$.

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

$x=-3+(1-(-3\left)\right)t$since $a=-3,b=5,c=1,d=4$

$$\begin{gathered} x=-3+(1+3)t \\ x=-3+4t \end{gathered}$$

Now take the points $(-3,5)$to $(1,4)$.

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

$$\begin{gathered} y=5+(4-5)t\text{since}a=-3,b=5,c=1,d=4 \\ y=5+(-1)t \\ y=5-t \end{gathered}$$

Thus, the parametric equations tor the line segment joining the pair of points $(-3,5),(1,4)$are $x=-3+4t,y=5-t,t\in [0,1]$.

Therefore, the parametric equations are $x=-3+4t,y=5-t,t\in [0,1].$