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 0.
So u • v = 0.
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 = 0 and the lengths.
We can make 0/25 simpler.
It 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
Does it make sense?
Let's draw u and v.
They meet at a square corner: 90°.
A dot product of 0 always means 90°.
It fits!
Let's look back.
First: find u • v.
Then: find the lengths.
Then: use the angle rule.
Then: find θ.
Last: check.