Angle between <1, 2> and <3, 1> Step by Step

Step by step

The question

Question: find the angle between u = ⟨1, 2⟩ and v = ⟨3, 1⟩.

u = ⟨1, 2⟩ and v = ⟨3, 1⟩

1 Find u • v

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

u • v = 1 × 3 + 2 × 1

1 times 3 is 3.

2 times 1 is 2.

u • v = 3 + 2

3 plus 2 is 5.

So u • v = 5.

u • v = 5

2 Find the lengths

The length of u: Pythagoras.

1 plus 4 is 5.

‖u‖ = √1² + 2² = √5

The length of v: Pythagoras.

9 plus 1 is 10.

‖v‖ = √3² + 1² = √10

3 Use the angle rule

The rule: cos θ is the dot product over the two lengths multiplied.

cos θ = u • v‖u‖ × ‖v‖

Now put the numbers in: u • v = 5 and the lengths.

cos θ = 5√5 × √10

Two roots multiply inside: √5 times √10 is √50.

√5 × √10 = √50

So the bottom is √50.

cos θ = 5√50

4 Find θ

With the calculator, that is about 0.7071.

5 ÷ √50 ≈ 0.7071

The cos⁻¹ button turns it back into the angle.

θ = cos⁻¹(0.7071)

Careful!

That number is cos θ, not the angle.

We still need cos⁻¹.

Careful: not θ = 0.7071 ✗

The calculator gives exactly 45.

So θ = 45°.

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!

u • v > 0 → θ < 90°

Let's look back.

First: find u • v.

Then: find the lengths.

Then: use the angle rule.

Then: find θ.

Last: check.

Answer