Step by step
The question
Question: find the angle between u = ⟨1, 2, 3⟩ and v = ⟨3, 2, 1⟩.
1 Find u • v
Multiply the matching parts: x times x, y times y, z times z.
Then add.
1 times 3 is 3.
2 times 2 is 4.
3 times 1 is 3.
Add them up: u • v = 10.
2 The two lengths
The length of u: square the parts and add.
1 plus 4 plus 9 is 14.
The length of v: square the parts and add.
9 plus 4 plus 1 is 14.
3 Find the angle
The rule: cos θ is the dot product over the two lengths multiplied.
Put in u • v = 10 and the two lengths.
Roots multiply inside: √196, which is 14.
10/14 is the same as 5/7.
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 about 44.415.
To the nearest tenth, θ ≈ 44.4°.
5 Check
The dot product is positive, so θ is less than 90°.
44.4° fits!
And cos of 44.4° is about 0.714, 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.