Angle between <5, 12> and <4, -3> Step by Step

Step by step

The question

Question: find the angle between u = ⟨5, 12⟩ and v = ⟨4, −3⟩.

u = ⟨5, 12⟩ and v = ⟨4, −3⟩

1 Find u • v

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

u • v = 5 × 4 + 12 × (−3)

5 times 4 is 20.

12 times (−3) is −36.

u • v = 20 − 36

20 minus 36 is −16.

So u • v = −16.

u • v = −16

2 Find the lengths

The length of u: Pythagoras.

25 plus 144 is 169.

The square root is 13.

‖u‖ = √5² + 12² = √169 = 13

The length of v: Pythagoras.

16 plus 9 is 25.

The square root is 5.

‖v‖ = √4² + (−3)² = √25 = 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 = −16 and the lengths.

cos θ = −1613 × 5

13 times 5 is 65.

So cos θ = −16/65.

cos θ = −1665

4 Find θ

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

θ = cos⁻¹(−1665)

Careful!

That number is cos θ, not the angle.

We still need cos⁻¹.

Careful: not θ = −16 over 65 ✗

The calculator gives about 104.25.

To the nearest tenth, θ ≈ 104.3°.

θ ≈ 104.3°

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°.

104.3° 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