Step by step
The question
Question: find the parametric equations of the line through (5, 1, 0) and (2, 3, 1).
(5, 1, 0) and (2, 3, 1)
1 Find the direction
Draw the three axes.
Start at (5, 1, 0).
The line also goes through (2, 3, 1).
The arrow from P to Q shows the direction.
The direction is end minus start, part by part.
v = ⟨2 − 5, 3 − 1, 1 − 0⟩
2 minus 5 is −3.
3 minus 1 is 2.
1 minus 0 is 1.
So v = ⟨−3, 2, 1⟩.
v = ⟨−3, 2, 1⟩
2 Write the equations
Start at the point, then add t times the direction.
t is any number: each t gives one point of the line.
r(t) = ⟨5, 1, 0⟩ + t⟨−3, 2, 1⟩
point + t × direction
Careful!
Only the direction is multiplied by t.
The start point stays as it is.
Careful: the point is added, the direction gets the t
Read each part: x = 5 − 3t.
x = 5 − 3t
Then: y = 1 + 2t.
y = 1 + 2t
And: z = t.
z = t
3 Check
Here is the whole line.
Let's check.
t = 0 gives (5, 1, 0), the start.
Right!
t = 0: (5 − 3(0), 1 + 2(0), (0)) = (5, 1, 0)
t = 1 gives (2, 3, 1), the other point.
Right!
t = 1: (2, 3, 1) = (2, 3, 1)
Let's look back.
First: find the direction.
Then: write the equations.
Last: check.
Answer
x = 5 − 3t