Chapter 10.4, Problem 7E

(a)

To determine

To find:The set of parametric equations.

Answer to Problem 7E

Theparametric equations are $x=t$ , $y=2t$ and $z=3t$ .

Explanation of Solution

Given information:The point is $\left(0,0,0\right)$ and vector is $v=\langle 1,2,3\rangle$ .

Formula used: The parametric equations for the line parallel to vector $v=\langle a,b,c\rangle$ and passing through the point $\left(x_1,y_1,z_1\right)$ are given by,

  $x=x_1+at$ , $y=y_1+bt$ and $z=z_1+ct$

Calculation:

The direction vector is $v=\langle 1,2,3\rangle$ . So, the direction numbers are $a=1$ , $b=2$ and $c=3$ .

The point is $\left(x_1,y_1,z_1\right)=\left(0,0,0\right)$ .

Substitute 0 for $x_1$ and $1$ for $a$ in the equation $x=x_1+at$ .

  $\begin{array}{c}x=0+1t\\ =t\end{array}$

Substitute 0 for $y_1$ and $2$ for $b$ in the equation $y=y_1+bt$ .

  $\begin{array}{c}y=0+2t\\ =2t\end{array}$

Substitute 0 for $z_1$ and $3$ for $c$ in the equation $z=z_1+ct$ .

  $\begin{array}{c}z=0+3t\\ =3t\end{array}$

Therefore, the parametric equations are $x=t$ , $y=2t$ and $z=3t$ .

(b)

To determine

To find:The set of symmetric equations for the line through the point $\left(0,0,0\right)$ and parallel to the specified vector $v=\langle 1,2,3\rangle$ .

Answer to Problem 7E

Theset of symmetric equation is $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ .

Explanation of Solution

Given information:The point is $\left(0,0,0\right)$ and vector is $v=\langle 1,2,3\rangle$ .

Calculation:

The direction numbers are all non-zero then symmetric equation can be calculated as follows:

  $\begin{array}{c}\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}\\ \frac{x-0}{1}=\frac{y-0}{2}=\frac{z-0}{3}\\ \frac{x}{1}=\frac{y}{2}=\frac{z}{3}\end{array}$

Therefore, the set of symmetric equation is $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ .