Symmetric equations of the line through (2, 1, 0) and (5, 3, 4) Step by Step

Step by step

The question

Question: find the symmetric equations of the line through (2, 1, 0) and (5, 3, 4).

(2, 1, 0) and (5, 3, 4)

1 Find the direction

Draw the three axes.

Start at (2, 1, 0).

The line also goes through (5, 3, 4).

The arrow from P to Q shows the direction.

The direction is end minus start, part by part.

v = ⟨5 − 2, 3 − 1, 4 − 0⟩

5 minus 2 is 3.

3 minus 1 is 2.

4 minus 0 is 4.

So v = ⟨3, 2, 4⟩.

v = ⟨3, 2, 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) = ⟨2, 1, 0⟩ + t⟨3, 2, 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 = 2 + 3t.

x = 2 + 3t

Then: y = 1 + 2t.

y = 1 + 2t

And: z = 4t.

z = 4t

3 Solve each for t

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

t = x − 23, t = y − 12, t = z4

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

= x − 23 = y − 12 = z4

4 Check

Here is the whole line.

Let's check.

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

Right!

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

t = 1 gives (5, 3, 4), the other point.

Right!

t = 1: (5, 3, 4) = (5, 3, 4)

Let's look back.

First: find the direction.

Then: write the equations.

Then: solve each for t.

Last: check.

Answer