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

Step by step

The question

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

u = ⟨1, 3⟩ and v = ⟨2, −1⟩

1 Find u • v

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

u • v = 1 × 2 + 3 × (−1)

1 times 2 is 2.

3 times (−1) is −3.

u • v = 2 − 3

2 minus 3 is −1.

So u • v = −1.

u • v = −1

2 Find the lengths

The length of u: Pythagoras.

1 plus 9 is 10.

‖u‖ = √1² + 3² = √10

The length of v: Pythagoras.

4 plus 1 is 5.

‖v‖ = √2² + (−1)² = √5

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√10 × √5

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

√10 × √5 = √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 98.13.

To the nearest tenth, θ ≈ 98.1°.

θ ≈ 98.1°

5 Check

Does it make sense?

Let's draw u and v.

The angle between them is a bit more than a square corner.

And the dot product is negative, so θ is more than 90°.

98.1° 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