Magnitude of <1, 2, 3>

Step by step

The question

Question: find the length of v = ⟨1, 2, 3⟩.

1 Draw it

Draw the three axes.

The arrow starts at the origin.

Go 1 along x, 2 along y, 3 along z.

The arrow goes from the origin to the end of that path.

2 Find the length

The length ‖v‖ is Pythagoras with three parts.

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

1 squared is 1.

2 squared is 4.

3 squared is 9.

Add: 1 plus 4 plus 9 is 14.

14 is not a square number.

So √14 is the exact length.

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

Careful!

We don't add the parts.

The arrow is the long side of a right triangle, so we square, add, and take the root.

3 Check

Let's check it as Pythagoras twice.

Across the floor the square is 5.

Then up z: 5 plus 9 is 14.

The same!

Let's look back.

First: draw it.

Then: find the length.

Last: check.