Steply

Line through (1, 0, -3) and (-2, 4, 1)

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, 0, −3) and (−2, 4, 1).

(1, 0, −3) and (−2, 4, 1)

1 Find the direction

Draw the three axes.

Start at (1, 0, −3).

The line also goes through (−2, 4, 1).

The arrow from P to Q shows the direction.

The direction is end minus start, part by part.

v = ⟨−2 − 1, 4 − 0, 1 − (−3)⟩

−2 minus 1 is −3.

4 minus 0 is 4.

1 minus (−3) is 4.

So v = ⟨−3, 4, 4⟩.

v = ⟨−3, 4, 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, 0, −3⟩ + t⟨−3, 4, 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 = 4t.

y = 4t

And: z = −3 + 4t.

z = −3 + 4t

3 Check

Here is the whole line.

Let's check.

t = 0 gives (1, 0, −3), the start.

Right!

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

t = 1 gives (−2, 4, 1), the other point.

Right!

t = 1: (−2, 4, 1) = (−2, 4, 1)

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 (1, 0, −1) and (0, 1, 3)
Show the answer
x = 1 − t, y = t, z = −1 + 4t

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