Video Transcript
Write down a set of parametric
equations for the straight line passing through the points 𝑃 sub zero three, three,
four and 𝑃 sub one five, two, negative four using 𝑃 sub one as your starting point
and the vector 𝐏 sub zero 𝐏 sub one as your direction vector.
Recall that parametric equations
for a line can be written down in the form 𝑥 equals 𝑥 sub zero plus 𝑙𝑡, 𝑦
equals 𝑦 sub zero plus 𝑚𝑡, and 𝑧 equals 𝑧 sub zero plus 𝑛𝑡 — here, 𝑥 sub
zero, 𝑦 sub zero, and 𝑧 sub zero the coordinates of any point on the line. The numbers 𝑙, 𝑚, and 𝑛 are the
components of any direction vector 𝐕 pointing in the direction of the line. 𝑡 is the parameter, which gives
the equations their name. It is thought of as running over
all real numbers.
Although we can choose any point on
the line for 𝑥 sub zero, 𝑦 sub zero, 𝑧 sub zero, we have been instructed to take
the point 𝑃 sub one as our starting point. So we take 𝑥 sub zero equals five,
𝑦 sub zero equals two, and 𝑧 sub zero equals negative four. We can calculate the direction
vector 𝐏 sub zero 𝐏 sub one as the difference of position vectors 𝐎𝐏 sub one
minus 𝐎𝐏 sub zero. So 𝐏 sub zero 𝐏 sub one equals
two, negative one, negative eight. So we have 𝑙 equals two, 𝑚 equals
negative one, and 𝑛 equals negative eight. Thus, a set of parametric equations
for this line is 𝑥 equals five plus two 𝑡, 𝑦 equals two minus 𝑡, and 𝑧 equals
negative four minus eight 𝑡.
It is worth emphasizing that
although we have written down one set of parametric equations for the line, this is
not the only choice. We could for example have started
with the point 𝑃 sub zero, resulting in the equations 𝑥 equals three plus two 𝑡,
𝑦 equals three minus 𝑡, 𝑧 equals four minus eight 𝑡.
Indeed, we can choose any point on
the line as our starting point, for example, the point halfway between 𝑃 sub zero
and 𝑃 sub one. As well as choosing any point on
the line as our starting point, we can choose any vector along the line as our
direction vector, that is, any positive or negative scalar multiple of 𝐕. All of these different sets of
parametric equations describe the same line.
