Step by step
The question
Question: find the parametric equations of the line through (0, 0, 0) and (2, 3, 1).
(0, 0, 0) and (2, 3, 1)
1 Find the direction
Draw the three axes.
Start at (0, 0, 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 − 0, 3 − 0, 1 − 0⟩
2 minus 0 is 2.
3 minus 0 is 3.
1 minus 0 is 1.
So v = ⟨2, 3, 1⟩.
v = ⟨2, 3, 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) = ⟨0, 0, 0⟩ + t⟨2, 3, 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 = 2t.
x = 2t
Then: y = 3t.
y = 3t
And: z = t.
z = t
3 Check
Here is the whole line.
Let's check.
t = 0 gives (0, 0, 0), the start.
Right!
t = 0: (2(0), 3(0), (0)) = (0, 0, 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 = 2t