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

Step by step

The question

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

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

1 Find the direction

Draw the three axes.

Start at (1, 4, 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 − 1, 1 − 4, 2 − 2⟩

3 minus 1 is 2.

1 minus 4 is −3.

2 minus 2 is 0.

So v = ⟨2, −3, 0⟩.

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

x = 1 + 2t

Then: y = 4 − 3t.

y = 4 − 3t

And: z = 2.

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

z = 2

3 Solve each for t

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

(z has no t: it stays as it is.)

t = x − 12, t = y − 4−3

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

And z stays 2.

= x − 12 = y − 4−3

4 Check

Here is the whole line.

Let's check.

t = 0 gives (1, 4, 2), the start.

Right!

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

Then: solve each for t.

Last: check.

Answer