Steply

Line through (2, 2, 2) and (-1, 4, 0)

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 (2, 2, 2) and (−1, 4, 0).

(2, 2, 2) and (−1, 4, 0)

1 Find the direction

Draw the three axes.

Start at (2, 2, 2).

The line also goes through (−1, 4, 0).

The arrow from P to Q shows the direction.

The direction is end minus start, part by part.

v = ⟨−1 − 2, 4 − 2, 0 − 2⟩

−1 minus 2 is −3.

4 minus 2 is 2.

0 minus 2 is −2.

So v = ⟨−3, 2, −2⟩.

v = ⟨−3, 2, −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, 2, 2⟩ + t⟨−3, 2, −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 − 3t.

x = 2 − 3t

Then: y = 2 + 2t.

y = 2 + 2t

And: z = 2 − 2t.

z = 2 − 2t

3 Check

Here is the whole line.

Let's check.

t = 0 gives (2, 2, 2), the start.

Right!

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

t = 1 gives (−1, 4, 0), the other point.

Right!

t = 1: (−1, 4, 0) = (−1, 4, 0)

Let's look back.

First: find the direction.

Then: write the equations.

Last: check.

Answer
x = 2 − 3t

Your turn

Try this one yourself, then check:

Find the parametric equations of the line through (3, 2, 4) and (1, 6, 2)
Show the answer
x = 3 − 2t, y = 2 + 4t, z = 4 − 2t

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