Step by step
The question
Question: find the parametric equations of the line through (0, 1, −2), parallel to ⟨3, 0, 1⟩.
(0, 1, −2) , ⟨3, 0, 1⟩
1 Find the direction
Draw the three axes.
Start at (0, 1, −2).
The direction ⟨3, 0, 1⟩ is given.
Parallel means the same direction.
So the direction is v = ⟨3, 0, 1⟩.
v = ⟨3, 0, 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, 1, −2⟩ + t⟨3, 0, 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 = 3t.
x = 3t
Then: y = 1.
The direction has no y part, so y stays 1.
y = 1
And: z = −2 + t.
z = −2 + t
3 Check
Here is the whole line.
Let's check.
t = 0 gives (0, 1, −2), the start.
Right!
t = 0: (3(0), 1, −2 + (0)) = (0, 1, −2)
t = 1 moves one whole direction: (3, 1, −1).
Right!
t = 1: (3, 1, −1) = (3, 1, −1)
Let's look back.
First: find the direction.
Then: write the equations.
Last: check.
Answer
x = 3t