Answered: Find sets of (a) parametric equations…

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Topics In Analytic Geometry

Section: Chapter Questions

Problem 12T

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Question

Find sets of (a) parametric equations and (b) symmetric equations of the line that passes through the two points (0, 0, 25), (10, 10, 0) (if possible). (For each line, write the direction numbers as integers.)

Expert Solution

Step 1

It is given that a line passing through two points $P (0, 0, 25)$ and $Q (10, 10, 0)$ .

Direction number of a line that passes through two points $(x_{1,} y_{1}, z_{1})$ and $(x_{2,} y_{2}, z_{2})$ is

$\begin{array}{rcl} l & = & x_{2} - x_{1}, m = y_{2} - y_{1}, n = z_{2} - z_{1} \\ l & = & 10 - 0, m = 10 - 0, n = 0 - 25 \end{array}$ $(w h e r e l, m, n a r e d i r e c t i o n n u m b e r s)$

Therefor,

$l = 10, m = 10, n = - 25$

$\vec{B} = 10 \hat{i} + 10 \hat{j} - 25 \hat{k}$ $(\vec{B} i s p a r a l l e l v e c t o r)$

Step 2

The parametric equation of line that passes through $(x_{1,} y_{1}, z_{1})$ and parallel to $\vec{B}$ is

$\overset{⇀}{r} = x_{1} \hat{i} + y_{1} \hat{j} + z_{1} \hat{k} + t (l \hat{i} + m \hat{j} + n \hat{k})$

$\begin{array}{rcl} \vec{r} & = & 0 \hat{i} + 0 \hat{j} + 25 \hat{k} + t (10 \hat{i} + 10 \hat{j} - 25 \hat{k}) \end{array}$

Now simplifying

$\vec{r} = 10 t \hat{i} + 10 t \hat{j} + (25 - 25 t) \hat{k}$

Hence the parametric equations are

$x = 10 t y = 10 t z = 25 - 25 t$

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