Worked examples
The question
Question: find the dot product of u = ⟨1, 2, 3⟩ and v = ⟨4, −5, 6⟩.
⟨1, 2, 3⟩ • ⟨4, −5, 6⟩
1 Multiply the matching parts
The dot product: multiply the matching parts: x times x, y times y, z times z.
Then add.
u • v = 1 × 4 + 2 × (−5) + 3 × 6
1 times 4 is 4.
2 times (−5) is −10.
3 times 6 is 18.
u • v = 4 − 10 + 18
Add them up: u • v = 12.
u • v = 12
Careful!
The dot product is not a vector.
We add the three products.
The answer is one number.
Careful: not ⟨4, −10, 18⟩ ✗
2 Check
Let's check: the other order, v dot u.
We get 12 again.
Right!
v • u = 4 × 1 + −5 × 2 + 6 × 3 = 12
A positive dot product means the arrows point in a similar way: the angle is less than 90°.
12 > 0 → angle less than 90°
Let's look back.
First: multiply the matching parts.
Last: check.
Answer Careful: not ⟨4, −10, 18⟩ ✗
▶ Watch it solved
The question
Question: find the dot product of u = ⟨−2, 5, 3⟩ and v = ⟨4, 1, −2⟩.
⟨−2, 5, 3⟩ • ⟨4, 1, −2⟩
1 Multiply the matching parts
The dot product: multiply the matching parts: x times x, y times y, z times z.
Then add.
u • v = −2 × 4 + 5 × 1 + 3 × (−2)
−2 times 4 is −8.
5 times 1 is 5.
3 times (−2) is −6.
u • v = −8 + 5 − 6
Add them up: u • v = −9.
u • v = −9
Careful!
The dot product is not a vector.
We add the three products.
The answer is one number.
Careful: not ⟨−8, 5, −6⟩ ✗
2 Check
Let's check: the other order, v dot u.
We get −9 again.
Right!
v • u = 4 × (−2) + 1 × 5 + −2 × 3 = −9
A negative dot product means the arrows point in opposite ways: the angle is more than 90°.
−9 < 0 → angle more than 90°
Let's look back.
First: multiply the matching parts.
Last: check.
Answer Careful: not ⟨−8, 5, −6⟩ ✗
▶ Watch it solved
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.
▶ Watch it solved
