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