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

Step by step

The question

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

u = ⟨2, −1, 2⟩ and v = ⟨−2, 2, 1⟩

1 Find u • v

Multiply the matching parts: x times x, y times y, z times z.

Then add.

u • v = 2 × (−2) + −1 × 2 + 2 × 1

2 times (−2) is −4.

−1 times 2 is −2.

2 times 1 is 2.

u • v = −4 − 2 + 2

Add them up: u • v = −4.

u • v = −4

2 The two lengths

The length of u: square the parts and add.

4 plus 1 plus 4 is 9.

The square root is 3.

‖u‖ = √2² + (−1)² + 2² = √9 = 3

The length of v: square the parts and add.

4 plus 4 plus 1 is 9.

The square root is 3.

‖v‖ = √(−2)² + 2² + 1² = √9 = 3

3 Find the angle

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

cos θ = u • v‖u‖ × ‖v‖

Put in u • v = −4 and the two lengths.

cos θ = −43 × 3

So cos θ = −4/9.

cos θ = −49

4 Find θ

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

θ = cos⁻¹(−49)

Careful!

That number is cos θ, not the angle.

We still need cos⁻¹.

Careful: not θ = −4/9 ✗

The calculator gives about 116.388.

To the nearest tenth, θ ≈ 116.4°.

θ ≈ 116.4°

5 Check

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

116.4° fits!

u • v < 0 → θ > 90°

And cos of 116.4° is about −0.444, the same number we had.

Right!

cos 116.4° ≈ −0.444

Let's look back.

First: find u • v.

Then: the two lengths.

Then: find the angle.

Then: find θ.

Last: check.

Answer