Vector equation of a line segment joining the points with position vectors \(\displaystyle{r}_{{0}}\) and \(\displaystyle{r}_{{1}}\) is

\(\displaystyle{r}={\left({1}-{t}\right)}{r}_{{0}}+{t}{r}_{{1}}\)

Where \(\displaystyle{t}\in{\left[{0},{1}\right]}\)

Substitute \(\displaystyle{r}_{{0}}={<}{0},-{1},{1}{>}\) and \(\displaystyle{r}_{{1}}={<}{\frac{{{1}}}{{{2}}}}.{\frac{{{1}}}{{{3}}}},{\frac{{{1}}}{{{4}}}}{>}\)

\(\displaystyle{r}{\left({t}\right)}={\left({1}-{t}\right)}{\left({0},-{1},{1}\right)}+{t}{<}{\frac{{{1}}}{{{2}}}},{\frac{{{1}}}{{{3}}}},{\frac{{{1}}}{{{4}}}}{>}\)

\(\displaystyle{r}{\left({t}\right)}={<}{0},-{1}+{t},{1}-{t}{>}+{<}{\frac{{{t}}}{{{2}}}},{\frac{{{t}}}{{{3}}}},{\frac{{{t}}}{{{4}}}}{>}\)

\(\displaystyle{r}{\left({t}\right)}={<}{\frac{{{t}}}{{{2}}}},-{1}+{\frac{{{4}{t}}}{{{3}}}},{1}-{\frac{{{3}{t}}}{{{4}}}}{>}\)

Where \(\displaystyle{t}\in{\left[{0},{1}\right]}\) The parametric equations for the line segment are

\(\displaystyle{x}={\frac{{{t}}}{{{2}}}},{y}=-{1}+{\frac{{{4}{t}}}{{{3}}}},{z}={1}-{\frac{{{3}{t}}}{{{4}}}}\)

Where \(\displaystyle{t}\in{\left[{0},{1}\right]}\)

\(\displaystyle{r}={\left({1}-{t}\right)}{r}_{{0}}+{t}{r}_{{1}}\)

Where \(\displaystyle{t}\in{\left[{0},{1}\right]}\)

Substitute \(\displaystyle{r}_{{0}}={<}{0},-{1},{1}{>}\) and \(\displaystyle{r}_{{1}}={<}{\frac{{{1}}}{{{2}}}}.{\frac{{{1}}}{{{3}}}},{\frac{{{1}}}{{{4}}}}{>}\)

\(\displaystyle{r}{\left({t}\right)}={\left({1}-{t}\right)}{\left({0},-{1},{1}\right)}+{t}{<}{\frac{{{1}}}{{{2}}}},{\frac{{{1}}}{{{3}}}},{\frac{{{1}}}{{{4}}}}{>}\)

\(\displaystyle{r}{\left({t}\right)}={<}{0},-{1}+{t},{1}-{t}{>}+{<}{\frac{{{t}}}{{{2}}}},{\frac{{{t}}}{{{3}}}},{\frac{{{t}}}{{{4}}}}{>}\)

\(\displaystyle{r}{\left({t}\right)}={<}{\frac{{{t}}}{{{2}}}},-{1}+{\frac{{{4}{t}}}{{{3}}}},{1}-{\frac{{{3}{t}}}{{{4}}}}{>}\)

Where \(\displaystyle{t}\in{\left[{0},{1}\right]}\) The parametric equations for the line segment are

\(\displaystyle{x}={\frac{{{t}}}{{{2}}}},{y}=-{1}+{\frac{{{4}{t}}}{{{3}}}},{z}={1}-{\frac{{{3}{t}}}{{{4}}}}\)

Where \(\displaystyle{t}\in{\left[{0},{1}\right]}\)