Question

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

Vectors
ANSWERED
asked 2021-08-15
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)

Expert Answers (1)

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