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