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Calculus / Calculus: Early Transcendentals 1 / Chapter 11.5 / Problem 9E

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

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Equations of lines Find equations of the

Chapter 13, Problem 9E

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QUESTION:

9-16. Equations of lines Find an equation of the following lines. Make a sketch of the line.

The line through (0, 0, 1) parallel to the y-axis

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QUESTION:

9-16. Equations of lines Find an equation of the following lines. Make a sketch of the line.

The line through (0, 0, 1) parallel to the y-axis

ANSWER:

Solution 9E

Step 1 of 3:

In this problem we need to find the equation of the line passing through the point P(0,0,1) and parallel to the y-axis.

Equation of a line :

An equation of the line passing through the point P( ${{x}_{0}}_{\ },{y}_{0}$ , ${{z}_{0}}_{\ }$ ) in the direction of the vector v = $\prec{}a\ ,\ b,\ c\succ{}$ is r = ${{r}_{0}}_{\ }$ + tv , here ${{r}_{0}}_{\ }$ = $\overline{OP}$ (or)

$\prec{}x,\ y\ ,z\succ{}$ = $\prec{}{{x}_{0}}_{\ },{y}_{0},{{z}_{0}}_{\ }\succ{}$ + t $\prec{}a\ ,\ b,\ c\succ{}$ , for - $\infty{}\ <\ t<\ \infty{}$ .

= $\prec{}{{x}_{0}}_{\ },{y}_{0},{{z}_{0}}_{\ }\succ{}$ + $\prec{}ta\ ,\ tb,\ tc\succ{}$

= $\prec{}{{x}_{0}+at}_{\ },{y}_{0}+bt,{{z}_{0}}_{\ }+ct\succ{}$

Therefore , the parametric equations of the line are ;

X = ${{x}_{0}}_{\ }$ + at , y = ${{y}_{0}}_{\ }$ +bt , and z = ${{z}_{0}}_{\ }$ + ct , for - $\infty{}\ <\ t<\ \infty{}$ .

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