Symmetric equations of the line through (0, 2, -1) parallel to <3… Step by Step

Step by step

The question

Question: find the symmetric equations of the line through (0, 2, −1), parallel to ⟨3, −1, 2⟩.

(0, 2, −1) , ⟨3, −1, 2⟩

1 Find the direction

Draw the three axes.

Start at (0, 2, −1).

The direction ⟨3, −1, 2⟩ is given.

Parallel means the same direction.

So the direction is v = ⟨3, −1, 2⟩.

v = ⟨3, −1, 2⟩

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, 2, −1⟩ + t⟨3, −1, 2⟩

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 = 2 − t.

y = 2 − t

And: z = −1 + 2t.

z = −1 + 2t

3 Solve each for t

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

t = x3, t = y − 2−1, t = z + 12

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

= x3 = y − 2−1 = z + 12

4 Check

Here is the whole line.

Let's check.

t = 0 gives (0, 2, −1), the start.

Right!

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

t = 1 moves one whole direction: (3, 1, 1).

Right!

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

Let's look back.

First: find the direction.

Then: write the equations.

Then: solve each for t.

Last: check.

Answer