Steply

Line through (1, 2, 3) and (4, 6, 8)

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, 2, 3) and (4, 6, 8).

(1, 2, 3) and (4, 6, 8)

1 Find the direction

Draw the three axes.

Start at (1, 2, 3).

The line also goes through (4, 6, 8).

The arrow from P to Q shows the direction.

The direction is end minus start, part by part.

v = ⟨4 − 1, 6 − 2, 8 − 3⟩

4 minus 1 is 3.

6 minus 2 is 4.

8 minus 3 is 5.

So v = ⟨3, 4, 5⟩.

v = ⟨3, 4, 5⟩

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, 2, 3⟩ + t⟨3, 4, 5⟩
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 = 2 + 4t.

y = 2 + 4t

And: z = 3 + 5t.

z = 3 + 5t

3 Check

Here is the whole line.

Let's check.

t = 0 gives (1, 2, 3), the start.

Right!

t = 0: (1 + 3(0), 2 + 4(0), 3 + 5(0)) = (1, 2, 3)

t = 1 gives (4, 6, 8), the other point.

Right!

t = 1: (4, 6, 8) = (4, 6, 8)

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 (5, 1, 5) and (9, 4, 8)
Show the answer
x = 5 + 4t, y = 1 + 3t, z = 5 + 3t

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