Video Transcript
Give the parametric equation of the
line that passes through the midpoint between 𝑝 one equals one, two, three and 𝑝
two equals three, six, negative five with direction vector negative one, two,
five.
We’re given that a line passes
through the midpoint between the two points 𝑝 one and 𝑝 two and has a direction
vector with components negative one, two, and five. Now, we know that the parametric
equations of a line in space are a nonunique set of three equations: 𝑥 equals 𝑥
sub 𝐴 plus 𝑡𝑙, 𝑦 equals 𝑦 sub 𝐴 plus 𝑡𝑚, and 𝑧 equals 𝑧 sub 𝐴 plus
𝑡𝑛. And that’s where 𝐴 is a point on
the line with coordinates 𝑥 sub 𝐴, 𝑦 sub 𝐴, and 𝑧 sub 𝐴; the vector with
components 𝑙, 𝑚, 𝑛 is a direction vector of the line; and 𝑡 is a real parameter
between negative and positive ∞.
We’ve been given the direction
vector for the line. So we can straightaway write down
the components 𝑙, 𝑚, and 𝑛. That’s 𝑙 equals negative one, 𝑚
equals two, and 𝑛 equals five. And to calculate a point on the
line, we can use the midpoint formula for the midpoint between 𝑝 one and 𝑝
two. This is just the point whose
coordinates are the averages of the 𝑥-, 𝑦-, and 𝑧-coordinates, respectively, of
the two given points. So, with 𝑝 one equals one, two,
three and 𝑝 two equals three, six, negative five, our midpoint has coordinates one
plus three over two, two plus six over two, and three plus negative five over
two. That’s four over two, eight over
two, and negative two over two, which is the point with coordinates two, four, and
negative one.
So now making some space, we have
our direction vector with components negative one, two, and five and the point on
the line with coordinates 𝑥 sub 𝐴 equals two, 𝑦 sub 𝐴 equals four, and 𝑧 sub 𝐴
equals negative one. And now substituting these into the
expressions for the parametric equations of the line, we have 𝑥 equals two, which
is 𝑥 sub 𝐴, plus negative one, which is 𝑙, times 𝑡. That’s 𝑥 equals two minus 𝑡. 𝑦 equals four, that’s 𝑦 sub 𝐴,
plus two, which is 𝑚, times 𝑡. And 𝑧 equals negative one, 𝑧 sub
𝐴, plus five, which is 𝑛, times 𝑡.
So the parametric equations of the
line passing through the midpoint of 𝑝 one and 𝑝 two with direction vector
negative one, two, five is 𝑥 equals two minus 𝑡, 𝑦 equals four plus two 𝑡, and
𝑧 equals negative one plus five 𝑡.
