Steply

Line through (0, 5, 2) and (3, 1, 2)

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 (0, 5, 2) and (3, 1, 2).

(0, 5, 2) and (3, 1, 2)

1 Find the direction

Draw the three axes.

Start at (0, 5, 2).

The line also goes through (3, 1, 2).

The arrow from P to Q shows the direction.

The direction is end minus start, part by part.

v = ⟨3 − 0, 1 − 5, 2 − 2⟩

3 minus 0 is 3.

1 minus 5 is −4.

2 minus 2 is 0.

So v = ⟨3, −4, 0⟩.

v = ⟨3, −4, 0⟩

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) = ⟨0, 5, 2⟩ + t⟨3, −4, 0⟩
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 = 3t.

x = 3t

Then: y = 5 − 4t.

y = 5 − 4t

And: z = 2.

The direction has no z part, so z stays 2.

z = 2

3 Check

Here is the whole line.

Let's check.

t = 0 gives (0, 5, 2), the start.

Right!

t = 0: (3(0), 5 − 4(0), 2) = (0, 5, 2)

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

Right!

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

Let's look back.

First: find the direction.

Then: write the equations.

Last: check.

Answer
x = 3t

Your turn

Try this one yourself, then check:

Find the parametric equations of the line through (0, 5, 5) and (3, 1, 5)
Show the answer
x = 3t, y = 5 − 4t, z = 5

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