Step by step
The question
Question: find the angle between u = ⟨6, 8⟩ and v = ⟨3, 4⟩.
1 Find u • v
The dot product: across times across, plus up times up.
6 times 3 is 18.
8 times 4 is 32.
18 plus 32 is 50.
So u • v = 50.
2 Find the lengths
The length of u: Pythagoras.
36 plus 64 is 100.
The square root is 10.
The length of v: Pythagoras.
9 plus 16 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 = 50 and the lengths.
We can make 50/50 simpler.
It is the same as 1.
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 0.
So θ = 0°.
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°.
0° fits!
Let's look back.
First: find u • v.
Then: find the lengths.
Then: use the angle rule.
Then: find θ.
Last: check.