Step by step
The question
Question: find the angle between u = ⟨3, 4⟩ and v = ⟨4, 3⟩.
1 Find u • v
The dot product: across times across, plus up times up.
3 times 4 is 12.
4 times 3 is 12.
12 plus 12 is 24.
So u • v = 24.
2 Find the lengths
The length of u: Pythagoras.
9 plus 16 is 25.
The square root is 5.
The length of v: Pythagoras.
16 plus 9 is 25.
The square root is 5.
3 Use the angle rule
The rule: cos θ is the dot product over the two lengths multiplied.
Now put the numbers in: u • v = 24 and the lengths.
5 times 5 is 25.
So cos θ = 24/25.
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 16.26.
To the nearest tenth, θ ≈ 16.3°.
5 Check
Does it make sense?
Let's draw u and v.
The arrows point almost the same way: a small angle.
And the dot product is positive, so θ is less than 90°.
16.3° fits!
Let's look back.
First: find u • v.
Then: find the lengths.
Then: use the angle rule.
Then: find θ.
Last: check.