Chapter 1.4, Problem 28E

To determine

To find: The parameterization for the ray that passes through the point $\left(-1,2\right)$ with initial point $\left(0,0\right)$ .

Answer to Problem 28E

A possible parameterization is $x=-1+t$ , and $y=2-2t$ for $t\ge 0$ .

Explanation of Solution

Given information: The initial point is $\left(-1,2\right)$ and the line passes through the point $\left(0,0\right)$ .

Calculation:

Substitute $-1$ for $x_0$ and $2$ for $y_0$ in the equation $x=x{0}_{}+at$ and $y=y{0}_{}+bt$ to find the parameterization of the line segment.

  $\begin{array}{c}x=-1+at\\ y=2+bt\end{array}$

The above equations represents a line that passes through $\left(-1,2\right)$ at $t=0$ .

To find the values of $a$ and $b$ assume the parameterization will pass through the point $\left(0,0\right)$ when $t=1$ . Substitute 0 for x and 0 for t in the equation $x=-1+at$ .

  $\begin{array}{c}0=-1+a\left(1\right)\\ 1=a\end{array}$

Substitute 0 for x and 0 for t in the equation $y=2+bt$ .

  $\begin{array}{c}0=2+b\left(1\right)\\ b=-2\end{array}$

Therefore, a possible parameterization is $x=-1+t$ , and $y=2-2t$ for $t\ge 0$ .