Question

Find a vector equation and parametric equations for the line segment that joins

Vectors
Find a vector equation and parametric equations for the line segment that joins P to Q.
P(0, - 1, 1), Q(1/2, 1/3, 1/4)

2021-08-16
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 $$\displaystyle{t}\in{\left[{0},{1}\right]}$$