Step by step
The question
Question: find the parametric equations of the line through (1, 2, 3), parallel to ⟨2, −1, 4⟩.
(1, 2, 3) , ⟨2, −1, 4⟩
1 Find the direction
Draw the three axes.
Start at (1, 2, 3).
The direction ⟨2, −1, 4⟩ is given.
Parallel means the same direction.
So the direction is v = ⟨2, −1, 4⟩.
v = ⟨2, −1, 4⟩
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) = ⟨1, 2, 3⟩ + t⟨2, −1, 4⟩
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 = 1 + 2t.
x = 1 + 2t
Then: y = 2 − t.
y = 2 − t
And: z = 3 + 4t.
z = 3 + 4t
3 Check
Here is the whole line.
Let's check.
t = 0 gives (1, 2, 3), the start.
Right!
t = 0: (1 + 2(0), 2 − (0), 3 + 4(0)) = (1, 2, 3)
t = 1 moves one whole direction: (3, 1, 7).
Right!
t = 1: (3, 1, 7) = (3, 1, 7)
Let's look back.
First: find the direction.
Then: write the equations.
Last: check.
Answer
x = 1 + 2t