Angle between <2, 2> and <3, 0>

Step by step

The question

Question: find the angle between u = ⟨2, 2⟩ and v = ⟨3, 0⟩.

1 Find u • v

The dot product: across times across, plus up times up.

2 times 3 is 6.

2 times 0 is 0.

6 plus 0 is 6.

So u • v = 6.

2 Find the lengths

The length of u: Pythagoras.

4 plus 4 is 8.

The length of v: Pythagoras.

9 plus 0 is 9.

The square root is 3.

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 = 6 and the lengths.

√8 times 3.

So the bottom is 3√8.

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

Does it make sense?

Let's draw u and v.

The angle between them is less than a square corner.

And the dot product is positive, so θ is less than 90°.

45° fits!

Let's look back.

First: find u • v.

Then: find the lengths.

Then: use the angle rule.

Then: find θ.

Last: check.