Steply

Line through (-1, 3, 2) and (2, 3, 6)

Step by step, with the reason for every step · Lines in space · Calculus 3

Step by step

The question

Question: find the parametric equations of the line through (−1, 3, 2) and (2, 3, 6).

(−1, 3, 2) and (2, 3, 6)

1 Find the direction

Draw the three axes.

Start at (−1, 3, 2).

The line also goes through (2, 3, 6).

The arrow from P to Q shows the direction.

The direction is end minus start, part by part.

v = ⟨2 − (−1), 3 − 3, 6 − 2⟩

2 minus (−1) is 3.

3 minus 3 is 0.

6 minus 2 is 4.

So v = ⟨3, 0, 4⟩.

v = ⟨3, 0, 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, 3, 2⟩ + t⟨3, 0, 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 + 3t.

x = −1 + 3t

Then: y = 3.

The direction has no y part, so y stays 3.

y = 3

And: z = 2 + 4t.

z = 2 + 4t

3 Check

Here is the whole line.

Let's check.

t = 0 gives (−1, 3, 2), the start.

Right!

t = 0: (−1 + 3(0), 3, 2 + 4(0)) = (−1, 3, 2)

t = 1 gives (2, 3, 6), the other point.

Right!

t = 1: (2, 3, 6) = (2, 3, 6)

Let's look back.

First: find the direction.

Then: write the equations.

Last: check.

Answer
x = −1 + 3t

Your turn

Try this one yourself, then check:

Find the parametric equations of the line through (−3, 1, 5) and (−1, 1, 9)
Show the answer
x = −3 + 2t, y = 1, z = 5 + 4t

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