# Equations of lines Find both the parametric and the vector equations of the following lines. The line through (0, 0, 1) in the direction of the vector v = <<4, 7, 0>>

Question
Equations
Equations of lines Find both the parametric and the vector equations of the following lines.
The line through (0, 0, 1) in the direction of the vector $$\displaystyle{v}={\left\langle{4},{7},{0}\right\rangle}$$

2021-01-20
Step 1
Given:
The line through (0, 0, 1) in the direction of the vector $$\displaystyle{v}={\left\langle{4},{7},{0}\right\rangle}$$.
To find:
Parametric and vector equations of the given line.
Step 2
Let,
$$\displaystyle\vec{{{r}_{{0}}}}={\left({0},{0},{1}\right)},\vec{{{d}}}={<}{4},{7},{0}{>}$$
The vector equation of the line is,
$$\displaystyle\vec{{{r}}}=\vec{{{r}_{{0}}}}+{t}\vec{{{d}}}$$
$$\displaystyle\Rightarrow\vec{{{r}}}={<}{0},{0},{1}{>}+{t}{<}{4},{7},{0}{>}$$
$$\displaystyle={<}{0},{0},{1}{>}+{<}{4}{t},{7}{t},{0}{>}$$
$$\displaystyle={<}{0}+{4}{t},{0}+{7}{t},{1}+{0}{>}$$
$$\displaystyle={<}{4}{t},{7}{t},{1}{>}$$
$$\displaystyle\Rightarrow\vec{{{r}}}={<}{4}{t},{7}{t},{0}{>}$$
The parametric equations of a line are,
Result:x = 4t, y = 7t, z = 0
Vector equation of a line is, $$\displaystyle\vec{{{r}}}={<}{4}{t},{7}{t},{0}{>}$$
Parametric equations of a line are,
x=4t, y = 7t, z =0.

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