Find sets of parametric equatons and symmetric equations of the

pamangking8 2021-11-30 Answered
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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Expert Answer

Momp1989
Answered 2021-12-01 Author has 8190 answers
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
\(\displaystyle{\frac{{{x}+{1}}}{{{3}}}}={\frac{{{y}}}{{{4}}}}={\frac{{{8}-{z}}}{{{8}}}}\)
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