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

Step by step

The question

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

u = ⟨1, 1, 1⟩ and v = ⟨1, −1, 2⟩

1 Find u • v

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

Then add.

u • v = 1 × 1 + 1 × (−1) + 1 × 2

1 times 1 is 1.

1 times (−1) is −1.

1 times 2 is 2.

u • v = 1 − 1 + 2

Add them up: u • v = 2.

u • v = 2

2 The two lengths

The length of u: square the parts and add.

1 plus 1 plus 1 is 3.

‖u‖ = √1² + 1² + 1² = √3

The length of v: square the parts and add.

1 plus 1 plus 4 is 6.

‖v‖ = √1² + (−1)² + 2² = √6

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 = 2 and the two lengths.

cos θ = 2√3 × √6

Two roots multiply inside: √18.

√3 × √6 = √18

So the bottom is √18.

cos θ = 2√18

4 Find θ

With the calculator, that is about 0.4714.

2 ÷ 4.243… ≈ 0.4714

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

θ = cos⁻¹(0.4714)

Careful!

That number is cos θ, not the angle.

We still need cos⁻¹.

Careful: not θ = 0.4714 ✗

The calculator gives about 61.874.

To the nearest tenth, θ ≈ 61.9°.

θ ≈ 61.9°

5 Check

The dot product is positive, so θ is less than 90°.

61.9° fits!

u • v > 0 → θ < 90°

And cos of 61.9° is about 0.471, the same number we had.

Right!

cos 61.9° ≈ 0.471

Let's look back.

First: find u • v.

Then: the two lengths.

Then: find the angle.

Then: find θ.

Last: check.

Answer