Angle between <3, 4> and <-6, -8> Step by Step

Step by step

The question

Question: find the angle between u = ⟨3, 4⟩ and v = ⟨−6, −8⟩.

u = ⟨3, 4⟩ and v = ⟨−6, −8⟩

1 Find u • v

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

u • v = 3 × (−6) + 4 × (−8)

3 times (−6) is −18.

4 times (−8) is −32.

u • v = −18 − 32

−18 minus 32 is −50.

So u • v = −50.

u • v = −50

2 Find the lengths

The length of u: Pythagoras.

9 plus 16 is 25.

The square root is 5.

‖u‖ = √3² + 4² = √25 = 5

The length of v: Pythagoras.

36 plus 64 is 100.

The square root is 10.

‖v‖ = √(−6)² + (−8)² = √100 = 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 = −50 and the lengths.

cos θ = −505 × 10

We can make −50/50 simpler.

It is the same as −1.

cos θ = −1

4 Find θ

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

θ = cos⁻¹(−1)

Careful!

That number is cos θ, not the angle.

We still need cos⁻¹.

Careful: not θ = −1 ✗

The calculator gives exactly 180.

So θ = 180°.

θ = 180°

5 Check

Does it make sense?

Let's draw u and v.

The arrows point almost opposite ways: a wide angle.

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

180° 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