Chapter 11.4, Problem 3E

To determine

To find: The appropriate words to fill in the blanks.

Answer to Problem 3E

Symmetric equation.

Explanation of Solution

Given information:

A statement is given as -

"When the direction number $a,b,c$ of the vector $\text{v}=\langle a,b,c\rangle$ all are non-zero, you can eliminate the parameter to obtain the _____ _______ of a line.”

Calculation:

Vectors are used for determining equation of a line in space. A line passing through the point $P(x_1,y_1,z_1)$ and parallel to the vector $\text{v}=\langle a,b,c\rangle$ is represented by the Parametric equations-

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

where, $a,b,c$ are direction numbers of vector $\text{v}=\langle a,b,c\rangle$ .

When direction numbers $a,b,c$ are non-zero then,

$\begin{array}{l}x=x_1+at\Rightarrow t=\frac{x-x_1}{a}\\ y=y_1+bt\Rightarrow t=\frac{y-y_1}{b}\\ z=z_1+ct\Rightarrow t=\frac{z-z_1}{c}\\ \text{Therefore, }\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}\end{array}$

This is called symmetric equation of a line.

Hence, appropriate words to fill in the blank would be −

Symmetric equation.