Angle between <3, 4, 0> and <0, 3, 4> Step by Step

Step by step

The question

Question: find the angle between u = ⟨3, 4, 0⟩ and v = ⟨0, 3, 4⟩.

u = ⟨3, 4, 0⟩ and v = ⟨0, 3, 4⟩

1 Find u • v

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

Then add.

u • v = 3 × 0 + 4 × 3 + 0 × 4

3 times 0 is 0.

4 times 3 is 12.

0 times 4 is 0.

u • v = 0 + 12 + 0

Add them up: u • v = 12.

u • v = 12

2 The two lengths

The length of u: square the parts and add.

9 plus 16 plus 0 is 25.

The square root is 5.

‖u‖ = √3² + 4² + 0² = √25 = 5

The length of v: square the parts and add.

0 plus 9 plus 16 is 25.

The square root is 5.

‖v‖ = √0² + 3² + 4² = √25 = 5

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 = 12 and the two lengths.

cos θ = 125 × 5

So cos θ = 12/25.

cos θ = 1225

4 Find θ

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

θ = cos⁻¹(1225)

Careful!

That number is cos θ, not the angle.

We still need cos⁻¹.

Careful: not θ = 12/25 ✗

The calculator gives about 61.315.

To the nearest tenth, θ ≈ 61.3°.

θ ≈ 61.3°

5 Check

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

61.3° fits!

u • v > 0 → θ < 90°

And cos of 61.3° is about 0.48, the same number we had.

Right!

cos 61.3° ≈ 0.48

Let's look back.

First: find u • v.

Then: the two lengths.

Then: find the angle.

Then: find θ.

Last: check.

Answer