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

Step by step

The question

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

u = ⟨2, 1⟩ and v = ⟨−1, 3⟩

1 Find u • v

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

u • v = 2 × (−1) + 1 × 3

2 times (−1) is −2.

1 times 3 is 3.

u • v = −2 + 3

−2 plus 3 is 1.

So u • v = 1.

u • v = 1

2 Find the lengths

The length of u: Pythagoras.

4 plus 1 is 5.

‖u‖ = √2² + 1² = √5

The length of v: Pythagoras.

1 plus 9 is 10.

‖v‖ = √(−1)² + 3² = √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 = 1 and the lengths.

cos θ = 1√5 × √10

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

√5 × √10 = √50

So the bottom is √50.

cos θ = 1√50

4 Find θ

With the calculator, that is about 0.1414.

1 ÷ √50 ≈ 0.1414

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

θ = cos⁻¹(0.1414)

Careful!

That number is cos θ, not the angle.

We still need cos⁻¹.

Careful: not θ = 0.1414 ✗

The calculator gives about 81.87.

To the nearest tenth, θ ≈ 81.9°.

θ ≈ 81.9°

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°.

81.9° 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