Step 1
Given:
The line through (0, 0, 1) in the direction of the vector\(v = <4, 7, 0>\).
To find:
Parametric and vector equations of the given line.
Step 2
Let,
\(\overrightarrow{r_0} = (0, 0, 1), \overrightarrow{d} = <4, 7, 0>\)
The vector equation of the line is,
\(\overrightarrow{r} = \overrightarrow{r_0} + t \overrightarrow{d}\)

\(\Rightarrow \overrightarrow{r} = <0, 0, 1> + t <4, 7, 0>\)

\(= <0, 0, 1> + <4t, 7t, 0>\)

\(= <0 + 4t, 0 + 7t, 1 + 0>\)

\(= <4t, 7t, 1>\) \(\Rightarrow \overrightarrow{r} = <4t, 7t, 0>\) The parametric equations of a line are, Result :\(x = 4t, y = 7t, z = 0\) Vector equation of a line is, \(\overrightarrow{r} = <4t, 7t, 0>\) Parametric equations of a line are, \(x = 4t, y = 7t, z = 0.\)

\(\Rightarrow \overrightarrow{r} = <0, 0, 1> + t <4, 7, 0>\)

\(= <0, 0, 1> + <4t, 7t, 0>\)

\(= <0 + 4t, 0 + 7t, 1 + 0>\)

\(= <4t, 7t, 1>\) \(\Rightarrow \overrightarrow{r} = <4t, 7t, 0>\) The parametric equations of a line are, Result :\(x = 4t, y = 7t, z = 0\) Vector equation of a line is, \(\overrightarrow{r} = <4t, 7t, 0>\) Parametric equations of a line are, \(x = 4t, y = 7t, z = 0.\)