Step by step
The question
Question: find the parametric equations of the line through (3, −2, 0), parallel to ⟨−1, 4, 2⟩.
(3, −2, 0) , ⟨−1, 4, 2⟩
1 Find the direction
Draw the three axes.
Start at (3, −2, 0).
The direction ⟨−1, 4, 2⟩ is given.
Parallel means the same direction.
So the direction is v = ⟨−1, 4, 2⟩.
v = ⟨−1, 4, 2⟩
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, 0⟩ + t⟨−1, 4, 2⟩
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 − t.
x = 3 − t
Then: y = −2 + 4t.
y = −2 + 4t
And: z = 2t.
z = 2t
3 Check
Here is the whole line.
Let's check.
t = 0 gives (3, −2, 0), the start.
Right!
t = 0: (3 − (0), −2 + 4(0), 2(0)) = (3, −2, 0)
t = 1 moves one whole direction: (2, 2, 2).
Right!
t = 1: (2, 2, 2) = (2, 2, 2)
Let's look back.
First: find the direction.
Then: write the equations.
Last: check.
Answer
x = 3 − t