Step by step
The question
Question: find the angle between u = ⟨1, 0, 0⟩ and v = ⟨1, 1, 0⟩.
1 Find u • v
Multiply the matching parts: x times x, y times y, z times z.
Then add.
1 times 1 is 1.
0 times 1 is 0.
0 times 0 is 0.
Add them up: u • v = 1.
2 The two lengths
The length of u: square the parts and add.
1 plus 0 plus 0 is 1.
The square root is 1.
The length of v: square the parts and add.
1 plus 1 plus 0 is 2.
3 Find the angle
The rule: cos θ is the dot product over the two lengths multiplied.
Put in u • v = 1 and the two lengths.
The bottom is 1√2.
So the bottom is 1√2.
4 Find θ
With the calculator, that is about 0.7071.
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 45.
So θ = 45°.
5 Check
The dot product is positive, so θ is less than 90°.
45° fits!
And cos of 45° is about 0.707, 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.