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.

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.