Step by step
The question
Question: find the parametric equations of the line through (1, 1, 1) and (3, 2, 5).
(1, 1, 1) and (3, 2, 5)
1 Find the direction
Draw the three axes.
Start at (1, 1, 1).
The line also goes through (3, 2, 5).
The arrow from P to Q shows the direction.
The direction is end minus start, part by part.
v = ⟨3 − 1, 2 − 1, 5 − 1⟩
3 minus 1 is 2.
2 minus 1 is 1.
5 minus 1 is 4.
So 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, 1, 1⟩ + 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 = 1 + t.
y = 1 + t
And: z = 1 + 4t.
z = 1 + 4t
3 Check
Here is the whole line.
Let's check.
t = 0 gives (1, 1, 1), the start.
Right!
t = 0: (1 + 2(0), 1 + (0), 1 + 4(0)) = (1, 1, 1)
t = 1 gives (3, 2, 5), the other point.
Right!
t = 1: (3, 2, 5) = (3, 2, 5)
Let's look back.
First: find the direction.
Then: write the equations.
Last: check.
Answer
x = 1 + 2t