Chapter 10.4, Problem 21E

To determine

To draw: The graph of the line if the equations are $x=3+2t$ , $y=2+t$ , and $z=1+\frac{1}{2}t$ .

Explanation of Solution

Given information:

The equations are $x=3+2t$ , $y=2+t$ , and $z=1+\frac{1}{2}t$ .

Graph:

The graph of the line with equation $x=3+2t$ , $y=2+t$ , $z=1+\frac{1}{2}t$ is shown below.

  

Interpretation:

To draw a line, two points are required. The two points can be calculated by considering two different value of $t$ .

Assume the value of $t$ as 2. Substitute the value of $t$ as 2 in the given equations.

  $\begin{array}{c}x=3+2\left(2\right)\\ =3+4\\ =7\end{array}$

and,

  $\begin{array}{c}y=2+2\\ =4\end{array}$

and,

  $\begin{array}{c}z=1+\frac{1}{2}\left(2\right)\\ =1+1\\ =2\end{array}$

From the above, the coordinate of the point is $\left(7,4,2\right)$ when the value of $t$ is 2.

Assume the value of $t$ as 4. Substitute the value of $t$ as 4 in the given equations.

  $\begin{array}{c}x=3+2\left(4\right)\\ =3+8\\ =11\end{array}$

and,

  $\begin{array}{c}y=2+4\\ =6\end{array}$

and,

  $\begin{array}{c}z=1+\frac{1}{2}\left(4\right)\\ =1+2\\ =3\end{array}$

From the above, the coordinate of the point is $\left(11,6,3\right)$ when the value of $t$ is 4.

Draw the graph of the line that passes through the point $\left(7,4,2\right)$ and $\left(11,6,3\right)$ .