# Find sets of parametric equatons and symmetric equations of the

Find sets of parametric equatons and symmetric equations of the line tha passes though the given point and is parallel to the given vector or line
$\begin{array}{|c|c|}\hline \text{Point} & \text{Parallel to} \\ \hline (-1,\ 0,\ 8) & v=3i+4j-8k \\ \hline \end{array}$
a) parametric equations
b) symmetric equations
$$\displaystyle{\frac{{{x}+{1}}}{{{3}}}}={\frac{{{y}}}{{{4}}}}={\frac{{{8}-{z}}}{{{8}}}}$$

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Momp1989
Step 1
Given:
The given point is $$\displaystyle{\left(−{1},\ {0},\ {8}\right)}$$ and the vector or line is $$\displaystyle{v}={3}{i}+{4}{j}-{8}{k}$$
To determine: The 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.
Step 2
a) The parametric equations for a line passing through $$\displaystyle{\left({x}_{{{0}}},\ {y}_{{{0}}},\ {z}_{{{0}}}\right)}$$ and parallel to the vector $$\displaystyle{v}={a}{i}+{b}{j}+{c}{k}$$ are
$$\displaystyle{x}={x}_{{{0}}}+{a}{t},\ {y}={y}_{{{0}}}+{b}{t},\ {z}={z}_{{{0}}}+{c}{t}$$
The required parametric equations are
$$\displaystyle{x}=-{1}+{3}{t},\ {y}={4}{t},\ {z}={8}-{8}{t}$$
Step 3
b) The parametric equations are
$$\displaystyle{x}=-{1}+{3}{t},\ {y}={4}{t},\ {z}={8}-{8}{t}$$
Solving for t we have,
$$\displaystyle\Rightarrow{t}={\frac{{{x}+{1}}}{{{3}}}},\ {t}={\frac{{{y}}}{{{4}}}},\ {t}={\frac{{{z}-{8}}}{{-{8}}}}$$
$$\displaystyle\Rightarrow{\frac{{{x}+{1}}}{{{3}}}}={\frac{{{y}}}{{{4}}}}={\frac{{{z}-{8}}}{{-{8}}}}$$
$$\displaystyle\Rightarrow{\frac{{{x}+{1}}}{{{3}}}}={\frac{{{y}}}{{{4}}}}={\frac{{{8}-{z}}}{{{8}}}}$$
Symmetric equations
$$\displaystyle{\frac{{{x}+{1}}}{{{3}}}}={\frac{{{y}}}{{{4}}}}={\frac{{{8}-{z}}}{{{8}}}}$$