Vector from (1, 1, 1) to (2, 3, 4) Step by Step

Step by step

The question

Question: find the vector from (1, 1, 1) to (2, 3, 4), and its length.

(1, 1, 1) to (2, 3, 4)

1 Draw it

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

The arrow starts at (1, 1, 1)

and ends at (2, 3, 4).

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 = ⟨2 − 1, 3 − 1, 4 − 1⟩

end − start

Careful!

Not start minus end.

That gives an arrow pointing the other way.

Careful: not ⟨−1, −2, −3⟩ ✗

2 minus 1 is 1.

3 minus 1 is 2.

4 minus 1 is 3.

So v = ⟨1, 2, 3⟩.

v = ⟨1, 2, 3⟩

3 Find the length

The length ‖v‖ is Pythagoras with three parts.

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

‖v‖ = √1² + 2² + 3²

1 squared is 1.

2 squared is 4.

3 squared is 9.

‖v‖ = √1 + 4 + 9

Add: 1 plus 4 plus 9 is 14.

‖v‖ = √14

14 is not a square number.

So √14 is the exact length.

‖v‖ = √14

With a calculator, that is about 3.7, to the nearest tenth.

‖v‖ ≈ 3.7

4 Check

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

We land on (2, 3, 4), the end!

(1 + 1, 1 + 2, 1 + 3) = (2, 3, 4)

And the length squared is 14, the sum of the squares.

Right!

(√14)² = 14

Let's look back.

First: draw it.

Then: work part by part.

Then: find the length.

Last: check.

Answer