Step by step
The question
Question: find the angle between u = ⟨1, 2, 2⟩ and v = ⟨2, 1, −2⟩.
1 Find u • v
Multiply the matching parts: x times x, y times y, z times z.
Then add.
1 times 2 is 2.
2 times 1 is 2.
2 times (−2) is −4.
Add them up: u • v = 0.
2 The two lengths
The length of u: square the parts and add.
1 plus 4 plus 4 is 9.
The square root is 3.
The length of v: square the parts and add.
4 plus 1 plus 4 is 9.
The square root is 3.
3 Find the angle
The rule: cos θ is the dot product over the two lengths multiplied.
Put in u • v = 0 and the two lengths.
0/9 is the same as 0.
4 Find θ
The cos⁻¹ button turns it back into the angle.
Careful!
That number is cos θ, not the angle.
We still need cos⁻¹.
The calculator gives exactly 90.
So θ = 90°.
5 Check
A dot product of 0 always means 90°.
It fits!
And cos of 90° is about 0, the same number we had.
Right!
Let's look back.
First: find u • v.
Then: the two lengths.
Then: find the angle.
Then: find θ.
Last: check.