Chapter 1.4, Problem 24E

To determine

To find: The parameterization for the line segment with endpoints $\left(-1,3\right)$ and $\left(3,-2\right)$ .

Answer to Problem 24E

A possible parameterization is $x=-1+4t$ and $y=3-5t$ for $0\le t\le 1$ .

Explanation of Solution

Given information: The endpoints are $\left(-1,3\right)$ and $\left(3,-2\right)$ .

Calculation:

Substitute $-1$ for $x_0$ and $3$ for $y_0$ in the equation $x=x{0}_{}+at$ and $y=y{0}_{}+bt$ to find the parameterization of the line segment.

  $\begin{array}{c}x=-1+at\\ y=3+bt\end{array}$

The above equations represents a line that passes through the point $\left(-1,3\right)$ at $t=0$ .

To find the values of $a$ and $b$ , assume the parameterization will pass through the point $\left(3,-2\right)$ for $t=1$ . Substitute 3 for $x$ and 1 for $t$ in the equation $x=-1+at$ .

  $\begin{array}{c}3=-1+a\left(1\right)\\ 3+1=a\\ a=4\end{array}$

Substitute $-2$ for $y$ and 1 for $t$ in the equation $y=-3+bt$ .

  $\begin{array}{c}-2=3+b\left(1\right)\\ b=-2-3\\ =-5\end{array}$

Therefore, a possible parameterization is $x=-1+4t$ and $y=3-5t$ for $0\le t\le 1$ .