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

Step by step

The question

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

(−2, 0, 1) , ⟨1, 2, −3⟩

1 Find the direction

Draw the three axes.

Start at (−2, 0, 1).

The direction ⟨1, 2, −3⟩ is given.

Parallel means the same direction.

So the direction is v = ⟨1, 2, −3⟩.

v = ⟨1, 2, −3⟩

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, 0, 1⟩ + t⟨1, 2, −3⟩

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 + t.

x = −2 + t

Then: y = 2t.

y = 2t

And: z = 1 − 3t.

z = 1 − 3t

3 Solve each for t

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

t = x + 2, t = y2, t = z − 1−3

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

= x + 2 = y2 = z − 1−3

4 Check

Here is the whole line.

Let's check.

t = 0 gives (−2, 0, 1), the start.

Right!

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

t = 1 moves one whole direction: (−1, 2, −2).

Right!

t = 1: (−1, 2, −2) = (−1, 2, −2)

Let's look back.

First: find the direction.

Then: write the equations.

Then: solve each for t.

Last: check.

Answer