Step by step
The question
Question: find the parametric equations of the line through (−3, 2, 4), parallel to ⟨2, 0, −1⟩.
(−3, 2, 4) , ⟨2, 0, −1⟩
1 Find the direction
Draw the three axes.
Start at (−3, 2, 4).
The direction ⟨2, 0, −1⟩ is given.
Parallel means the same direction.
So the direction is v = ⟨2, 0, −1⟩.
v = ⟨2, 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) = ⟨−3, 2, 4⟩ + t⟨2, 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 = −3 + 2t.
x = −3 + 2t
Then: y = 2.
The direction has no y part, so y stays 2.
y = 2
And: z = 4 − t.
z = 4 − t
3 Check
Here is the whole line.
Let's check.
t = 0 gives (−3, 2, 4), the start.
Right!
t = 0: (−3 + 2(0), 2, 4 − (0)) = (−3, 2, 4)
t = 1 moves one whole direction: (−1, 2, 3).
Right!
t = 1: (−1, 2, 3) = (−1, 2, 3)
Let's look back.
First: find the direction.
Then: write the equations.
Last: check.
Answer
x = −3 + 2t