Steply

Line through (2, -1, 4) and (5, 1, 3)

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, −1, 4) and (5, 1, 3).

(2, −1, 4) and (5, 1, 3)

1 Find the direction

Draw the three axes.

Start at (2, −1, 4).

The line also goes through (5, 1, 3).

The arrow from P to Q shows the direction.

The direction is end minus start, part by part.

v = ⟨5 − 2, 1 − (−1), 3 − 4⟩

5 minus 2 is 3.

1 minus (−1) is 2.

3 minus 4 is −1.

So v = ⟨3, 2, −1⟩.

v = ⟨3, 2, −1⟩

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, −1, 4⟩ + t⟨3, 2, −1⟩
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 = −1 + 2t.

y = −1 + 2t

And: z = 4 − t.

z = 4 − t

3 Check

Here is the whole line.

Let's check.

t = 0 gives (2, −1, 4), the start.

Right!

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

t = 1 gives (5, 1, 3), the other point.

Right!

t = 1: (5, 1, 3) = (5, 1, 3)

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 (2, −5, 2) and (3, −1, −1)
Show the answer
x = 2 + t, y = −5 + 4t, z = 2 − 3t

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