Symmetric equations of the line through (1, -2, 3) and (3, 2, 4) Step by Step

Step by step

The question

Question: find the symmetric equations of the line through (1, −2, 3) and (3, 2, 4).

(1, −2, 3) and (3, 2, 4)

1 Find the direction

Draw the three axes.

Start at (1, −2, 3).

The line also goes through (3, 2, 4).

The arrow from P to Q shows the direction.

The direction is end minus start, part by part.

v = ⟨3 − 1, 2 − (−2), 4 − 3⟩

3 minus 1 is 2.

2 minus (−2) is 4.

4 minus 3 is 1.

So v = ⟨2, 4, 1⟩.

v = ⟨2, 4, 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) = ⟨1, −2, 3⟩ + t⟨2, 4, 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 = 1 + 2t.

x = 1 + 2t

Then: y = −2 + 4t.

y = −2 + 4t

And: z = 3 + t.

z = 3 + t

3 Solve each for t

Solve each equation for t: take away the start number, then divide by the direction number.

t = x − 12, t = y + 24, t = z − 3

All are the same t, so they are equal to each other: the symmetric equations.

= x − 12 = y + 24 = z − 3

4 Check

Here is the whole line.

Let's check.

t = 0 gives (1, −2, 3), the start.

Right!

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

t = 1 gives (3, 2, 4), the other point.

Right!

t = 1: (3, 2, 4) = (3, 2, 4)

Let's look back.

First: find the direction.

Then: write the equations.

Then: solve each for t.

Last: check.

Answer