Worked examples
The question
Question: find the vector from (1, 2, 3) to (4, 6, 3), and its length.
(1, 2, 3) to (4, 6, 3)
1 Draw it
Draw the three axes: x toward us, y right, z up.
The arrow starts at (1, 2, 3)
and ends at (4, 6, 3).
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 = ⟨4 − 1, 6 − 2, 3 − 3⟩
end − start
Careful!
Not start minus end.
That gives an arrow pointing the other way.
Careful: not ⟨−3, −4, 0⟩ ✗
4 minus 1 is 3.
6 minus 2 is 4.
3 minus 3 is 0.
So v = ⟨3, 4, 0⟩.
v = ⟨3, 4, 0⟩
3 Find the length
The length ‖v‖ is Pythagoras with three parts.
Square each part, add, then take the square root.
‖v‖ = √32 + 42 + 02
3 squared is 9.
4 squared is 16.
0 squared is 0.
‖v‖ = √9 + 16 + 0
Add: 9 plus 16 plus 0 is 25.
‖v‖ = √25
5 times 5 is 25.
5 × 5 = 25
So ‖v‖ = 5.
‖v‖ = 5
4 Check
Let's check: start at (1, 2, 3) and add the parts.
We land on (4, 6, 3), the end!
(1 + 3, 2 + 4, 3 + 0) = (4, 6, 3)
And the length squared is 25, the sum of the squares.
Right!
5² = 25
Let's look back.
First: draw it.
Then: work part by part.
Then: find the length.
Last: check.
▶ Watch it solved
The question
Question: find the vector from (2, −1, 4) to (4, 1, 5), and its length.
(2, −1, 4) to (4, 1, 5)
1 Draw it
Draw the three axes: x toward us, y right, z up.
The arrow starts at (2, −1, 4)
and ends at (4, 1, 5).
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 = ⟨4 − 2, 1 − (−1), 5 − 4⟩
end − start
Careful!
Not start minus end.
That gives an arrow pointing the other way.
Careful: not ⟨−2, −2, −1⟩ ✗
4 minus 2 is 2.
1 minus (−1) is 2.
5 minus 4 is 1.
So v = ⟨2, 2, 1⟩.
v = ⟨2, 2, 1⟩
3 Find the length
The length ‖v‖ is Pythagoras with three parts.
Square each part, add, then take the square root.
‖v‖ = √22 + 22 + 12
2 squared is 4.
2 squared is 4.
1 squared is 1.
‖v‖ = √4 + 4 + 1
Add: 4 plus 4 plus 1 is 9.
‖v‖ = √9
3 times 3 is 9.
3 × 3 = 9
So ‖v‖ = 3.
‖v‖ = 3
4 Check
Let's check: start at (2, −1, 4) and add the parts.
We land on (4, 1, 5), the end!
(2 + 2, −1 + 2, 4 + 1) = (4, 1, 5)
And the length squared is 9, the sum of the squares.
Right!
3² = 9
Let's look back.
First: draw it.
Then: work part by part.
Then: find the length.
Last: check.
▶ Watch it solved
The question
Question: find the length of v = ⟨1, 2, 3⟩.
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.
‖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
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‖ = 1 + 2 + 3 ✗
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!
floor: 1 + 4 = 5, then 5 + 9 = 14
Let's look back.
First: draw it.
Then: find the length.
Last: check.
Answer Careful: not ‖v‖ = 1 + 2 + 3 ✗
▶ Watch it solved
