The question
Question: find the dot product of u = ⟨3, 4⟩ and v = ⟨2, 1⟩.
u = ⟨3, 4⟩ and v = ⟨2, 1⟩
1 Multiply the matching parts
The dot product: across times across, plus up times up.
u • v = 3 × 2 + 4 × 1
3 times 2 is 6.
4 times 1 is 4.
u • v = 6 + 4
6 plus 4 is 10.
So u • v = 10.
u • v = 10
Careful!
The dot product is not a vector.
We add the two products.
The answer is one number.
Careful: not ⟨6, 4⟩ ✗
2 Check
Does it make sense?
Let's draw u and v.
The angle between them is less than a square corner.
A positive dot product means the angle is less than 90°.
10 is positive.
It fits!
10 > 0 → less than 90°
Let's look back.
First: multiply the matching parts.
Last: check.
Answer Careful: not ⟨6, 4⟩ ✗
▶ Watch it solved
The question
Question: find the dot product of u = ⟨−2, 3⟩ and v = ⟨4, 1⟩.
u = ⟨−2, 3⟩ and v = ⟨4, 1⟩
1 Multiply the matching parts
The dot product: across times across, plus up times up.
u • v = −2 × 4 + 3 × 1
−2 times 4 is −8.
3 times 1 is 3.
u • v = −8 + 3
−8 plus 3 is −5.
So u • v = −5.
u • v = −5
Careful!
The dot product is not a vector.
We add the two products.
The answer is one number.
Careful: not ⟨−8, 3⟩ ✗
2 Check
Does it make sense?
Let's draw u and v.
The angle between them is more than a square corner.
A negative dot product means the angle is more than 90°.
−5 is negative.
It fits!
−5 < 0 → more than 90°
Let's look back.
First: multiply the matching parts.
Last: check.
Answer Careful: not ⟨−8, 3⟩ ✗
▶ Watch it solved
The question
Question: find the angle between u = ⟨1, 2⟩ and v = ⟨3, 1⟩.
u = ⟨1, 2⟩ and v = ⟨3, 1⟩
1 Find u • v
The dot product: across times across, plus up times up.
u • v = 1 × 3 + 2 × 1
1 times 3 is 3.
2 times 1 is 2.
u • v = 3 + 2
3 plus 2 is 5.
So u • v = 5.
u • v = 5
2 Find the lengths
The length of u: Pythagoras.
1 plus 4 is 5.
‖u‖ = √12 + 22 = √5
The length of v: Pythagoras.
9 plus 1 is 10.
‖v‖ = √32 + 12 = √10
3 Use the angle rule
The rule: cos θ is the dot product over the two lengths multiplied.
cos θ = u • v‖u‖ × ‖v‖
Now put the numbers in: u • v = 5 and the lengths.
cos θ = 5√5 × √10
Two roots multiply inside: √5 times √10 is √50.
√5 × √10 = √50
So the bottom is √50.
cos θ = 5√50
4 Find θ
With the calculator, that is about 0.7071.
5 ÷ √50 ≈ 0.7071
The cos⁻¹ button turns it back into the angle.
θ = cos−1(0.7071)
Careful!
That number is cos θ, not the angle.
We still need cos⁻¹.
Careful: not θ = 0.7071 ✗
The calculator gives exactly 45.
So θ = 45°.
θ = 45°
5 Check
Does it make sense?
Let's draw u and v.
The angle between them is less than a square corner.
And the dot product is positive, so θ is less than 90°.
45° fits!
u • v > 0 → θ < 90°
Let's look back.
First: find u • v.
Then: find the lengths.
Then: use the angle rule.
Then: find θ.
Last: check.
▶ Watch it solved
