Angle between <1, -1, 0> and <-1, 0, 1> Step by Step

Step by step

The question

Question: find the angle between u = ⟨1, −1, 0⟩ and v = ⟨−1, 0, 1⟩.

u = ⟨1, −1, 0⟩ and v = ⟨−1, 0, 1⟩

1 Find u • v

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

Then add.

u • v = 1 × (−1) + −1 × 0 + 0 × 1

1 times (−1) is −1.

−1 times 0 is 0.

0 times 1 is 0.

u • v = −1 + 0 + 0

Add them up: u • v = −1.

u • v = −1

2 The two lengths

The length of u: square the parts and add.

1 plus 1 plus 0 is 2.

‖u‖ = √1² + (−1)² + 0² = √2

The length of v: square the parts and add.

1 plus 0 plus 1 is 2.

‖v‖ = √(−1)² + 0² + 1² = √2

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 = −1 and the two lengths.

cos θ = −1√2 × √2

Roots multiply inside: √4, which is 2.

√2 × √2 = √4 = 2

So cos θ = −1/2.

cos θ = −12

4 Find θ

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

θ = cos⁻¹(−12)

Careful!

That number is cos θ, not the angle.

We still need cos⁻¹.

Careful: not θ = −1/2 ✗

The calculator gives exactly 120.

So θ = 120°.

θ = 120°

5 Check

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

120° fits!

u • v < 0 → θ > 90°

And cos of 120° is about −0.5, the same number we had.

Right!

cos 120° ≈ −0.5

Let's look back.

First: find u • v.

Then: the two lengths.

Then: find the angle.

Then: find θ.

Last: check.

Answer