# Find sets of parametric equations and symmetric equations of the

Find sets of parametric equations and symmetric equations of the line that passes through the given point and is parallel to the given vector or line.
$\begin{array}{|c|c|}\hline \text{Point} & \text{Parallel to}v= \\ \hline (0,\ 0,\ 0) & <<8,\ 1,\ 4>> \\ \hline \end{array}$
(a) parametric equations
(b) symmetric equations
$$\displaystyle{8}{x}={y}={4}{z}$$
$$\displaystyle{4}{x}={y}={8}{z}$$
$$\displaystyle{\frac{{{x}}}{{{8}}}}={y}={\frac{{{z}}}{{{4}}}}$$
$$\displaystyle{\frac{{{x}}}{{{4}}}}={y}={\frac{{{z}}}{{{8}}}}$$

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Step 1
consider the t equation
Point $$\displaystyle={\left({0},\ {0},\ {0}\right)}$$
$$\displaystyle{v}={\left({8},\ {1},\ {4}\right)}$$
Step 2
Sets of parametric equations and symmetric equations of the line that passes through the given point
Point $$\displaystyle={\left({0},\ {0},\ {0}\right)}$$
$$\displaystyle{v}={\left({8},\ {1},\ {4}\right)}$$
To find a set of parametric equations of the line, we use the coordinates:
$$\displaystyle{x}_{{{1}}}={0}$$
$$\displaystyle{y}_{{{1}}}={0}$$
$$\displaystyle{z}_{{{1}}}={0}$$
The direction numbers:
$$\displaystyle{a}={8},\ {b}={1},\ {c}={4}$$
The parametric equations are:
$$\displaystyle{x}={0}+{8}={8}$$
$$\displaystyle{y}={0}+{1}={1}$$
$$\displaystyle{z}={0}+{4}={4}$$
(b) symmetric equation for
$$\displaystyle{8}{x}={y}={4}{z}$$
$$\displaystyle{\frac{{{x}-{x}_{{{1}}}}}{{{a}}}}={\frac{{{y}-{y}_{{{1}}}}}{{{b}}}}={\frac{{{z}-{z}_{{{1}}}}}{{{c}}}}$$
Substitute $$\displaystyle{\left({x},\ {y},\ {z}\right)}={\left({0},\ {0},\ {0}\right)}{n}={\left\langle{a},\ {b},\ {c}\right\rangle}={\left\langle{8},\ {1},\ {4}\right\rangle}$$
in symmetric equation
$$\displaystyle{\frac{{{x}-{0}}}{{{8}}}}={\frac{{{y}-{0}}}{{{1}}}}={\frac{{{z}-{3}}}{{{4}}}}$$
$$\displaystyle{\frac{{{8}-{8}}}{{{0}}}}={\frac{{{y}-{0}}}{{{1}}}}={\frac{{{4}-{3}}}{{{4}}}}$$
$$\displaystyle{\frac{{{8}}}{{{8}}}}={\frac{{{y}-{0}}}{{{1}}}}={\frac{{{1}}}{{{4}}}}$$
Direction number as integers $$\displaystyle{1},\ {1},\ {4}.$$
symmetric equation for $$\displaystyle{4}{x}={y}-{8}{Z}$$
substitute $$\displaystyle{\left({x},\ {y},\ {z}\right)}={\left({0},\ {0},\ {0}\right)}{n}={\left\langle{a},\ {b},\ {c}\right\rangle}={\left\langle{8},\ {1},\ {4}\right\rangle}{\frac{{{x}-{0}}}{{{8}}}}={\frac{{{y}-{0}}}{{{1}}}}={\frac{{{z}-{3}}}{{{4}}}}$$
in symmetric equation $$\displaystyle{\frac{{{0}-{8}}}{{{8}}}}={\frac{{{0}-{0}}}{{{1}}}}={\frac{{{4}-{3}}}{{{4}}}}$$
$$\displaystyle{\frac{{{4}-{0}}}{{{8}}}}={\frac{{{0}-{0}}}{{{1}}}}={\frac{{{8}-{4}}}{{{4}}}}$$ direction number as integer 1, 1, 4
$$\displaystyle{\frac{{{4}}}{{{8}}}}={\frac{{{0}}}{{{1}}}}={\frac{{{4}}}{{{4}}}}$$
Direction number as integer 2, 1, 1