Chapter 12, Problem 23CLT

a.

To determine

Find parametric equations of the line passing through the points.

Answer to Problem 23CLT

  $\boxed{\begin{array}{l}x=-2+7t\\ y=3+5t\\ z=25t\end{array}}$

Explanation of Solution

Given information:

Find sets of parametric equations of the line passing through the points $\left(-2,3,0\right)$ and $\left(5,8,25\right)$ .

Calculation:

Consider the parametric equations of the line passing through the points

  $\left(-2,3,0\right)$ & $\left(5,8,25\right)$

  $\left(x_1,y_1,z_1\right)$ and $\left(x_2,y_2,z_2\right)$ is,

  $\begin{array}{l}x=x_1+t\left(x_2-x_1\right)\\ y=y_1+t\left(y_2-y_1\right)\\ z=z_1+t\left(z_2-z_1\right)\end{array}$

Now substitute the given values in above expression,

  $\begin{array}{l}x=-2+t\left(5+2\right)\\ x=-2+7t\\ y=3+t\left(8-3\right)\\ y=3+5t\\ z=0+t\left(25-0\right)\\ z=25t\end{array}$

Hence the parametric equations of the line passing through the points $\left(-2,3,0\right)$ & $\left(5,8,25\right)$ is

  $\boxed{\begin{array}{l}x=-2+7t\\ y=3+5t\\ z=25t\end{array}}$

b.

To determine

Find symmetric equations of the line passing through the points.

Answer to Problem 23CLT

  $\boxed{\frac{x+2}{7},\frac{y-3}{5},\frac{z}{25}}$

Explanation of Solution

Given information:

Find sets of symmetric equations of the line passing through the points $\left(-2,3,0\right)$ and $\left(5,8,25\right)$ .

Calculation:

Consider the parametric equations of the line passing through the points

  $\left(-2,3,0\right)$ & $\left(5,8,25\right)$

  $\left(x_1,y_1,z_1\right)$ and $\left(x_2,y_2,z_2\right)$ is,

  $\begin{array}{l}x=x_1+t\left(x_2-x_1\right)\\ y=y_1+t\left(y_2-y_1\right)\\ z=z_1+t\left(z_2-z_1\right)\end{array}$

Now substitute the given values in above expression,

  $\begin{array}{l}x=-2+t\left(5+2\right)\\ x=-2+7t\\ y=3+t\left(8-3\right)\\ y=3+5t\\ z=0+t\left(25-0\right)\\ z=25t\end{array}$

Express $t$ by equating three symmetric equations are.

  $\begin{array}{l}x=-2+t\left(5+2\right)\\ x=-2+7t\\ t=\frac{x+2}{7}\\ y=3+5t\\ t=\frac{y-3}{5}\\ z=25t\\ t=\frac{z}{25}\end{array}$

Hence the symmetrical equations are symmetrical equations

  $\boxed{\frac{x+2}{7},\frac{y-3}{5},\frac{z}{25}}$