Vector from (1, 0, -2) to (3, 6, 1) Step by Step

Step by step

The question

Question: find the vector from (1, 0, −2) to (3, 6, 1), and its length.

(1, 0, −2) to (3, 6, 1)

1 Draw it

Draw the three axes: x toward us, y right, z up.

The arrow starts at (1, 0, −2)

and ends at (3, 6, 1).

Draw the arrow from the start to the end.

We call it v.

2 Work part by part

Each part is end minus start: x minus x, y minus y, z minus z.

v = ⟨3 − 1, 6 − 0, 1 − (−2)⟩

end − start

Careful!

Not start minus end.

That gives an arrow pointing the other way.

Careful: not ⟨−2, −6, −3⟩ ✗

3 minus 1 is 2.

6 minus 0 is 6.

1 minus (−2) is 3.

So v = ⟨2, 6, 3⟩.

v = ⟨2, 6, 3⟩

3 Find the length

The length ‖v‖ is Pythagoras with three parts.

Square each part, add, then take the square root.

‖v‖ = √2² + 6² + 3²

2 squared is 4.

6 squared is 36.

3 squared is 9.

‖v‖ = √4 + 36 + 9

Add: 4 plus 36 plus 9 is 49.

‖v‖ = √49

7 times 7 is 49.

7 × 7 = 49

So ‖v‖ = 7.

‖v‖ = 7

4 Check

Let's check: start at (1, 0, −2) and add the parts.

We land on (3, 6, 1), the end!

(1 + 2, 0 + 6, −2 + 3) = (3, 6, 1)

And the length squared is 49, the sum of the squares.

Right!

7² = 49

Let's look back.

First: draw it.

Then: work part by part.

Then: find the length.

Last: check.

Answer