Chapter 1, Problem 49RE

To determine

To find: The parametrization of the line segment with endpoints $\left(-2,5\right)$ and $\left(4,3\right)$ .

Answer to Problem 49RE

The parametrization is $x=-2+6t$ and $y=5-2t$ for $0\le t\le 1$ .

Explanation of Solution

Given information:The linesegment with endpoints $\left(-2,5\right)$ and $\left(4,3\right)$ .

Calculation:

A possible parametrization of a line segment can be found using $x=x_0+at$ and $y=y_0+bt$ .

The parametrization will pass through the point $\left(-2,5\right)$ for $t=0$ and the other endpoint of $\left(4,3\right)$ for $t=1$ .

Substitute $-2$ for $x_0$ and 4 for xin the equation $x=x_0+at$ .

  $4=-2+at$

Substitute 5 for $x_0$ and 3 for y in the equation $y=y_0+bt$ .

  $3=5+bt$

For $t=0$ ,

  $\begin{array}{c}4=-2+a\left(1\right)\\ a=4+2\\ =6\end{array}$

For $t=1$ ,

Substitute the value of $x$ , $y$ and $t$ into the equation.

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

Therefore, the parametrization is $x=-2+6t$ and $y=5-2t$ for $0\le t\le 1$ .