Step by step
The question
Question: find the length of v = ⟨−2, 6, 9⟩.
v = ⟨−2, 6, 9⟩
1 Draw it
Draw the three axes.
The arrow starts at the origin.
Go −2 along x, 6 along y, 9 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.
‖v‖ = √(−2)2 + 62 + 92
(−2) squared is 4.
6 squared is 36.
9 squared is 81.
Minus times minus is plus.
‖v‖ = √4 + 36 + 81
Add: 4 plus 36 plus 81 is 121.
‖v‖ = √121
11 times 11 is 121.
11 × 11 = 121
So ‖v‖ = 11.
‖v‖ = 11
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.
Careful: not ‖v‖ = −2 + 6 + 9 ✗
3 Check
Let's check it as Pythagoras twice.
Across the floor the square is 40.
Then up z: 40 plus 81 is 121.
The same!
floor: 4 + 36 = 40, then 40 + 81 = 121
Let's look back.
First: draw it.
Then: find the length.
Last: check.
Answer
Careful: not ‖v‖ = −2 + 6 + 9 ✗