Step by step
The question
Question: find the dot product of u = ⟨3, −2⟩ and v = ⟨−1, −4⟩.
u = ⟨3, −2⟩ and v = ⟨−1, −4⟩
1 Multiply the matching parts
The dot product: across times across, plus up times up.
u • v = 3 × (−1) + −2 × (−4)
3 times (−1) is −3.
−2 times (−4) is 8.
u • v = −3 + 8
−3 plus 8 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 ⟨−3, 8⟩ ✗
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°.
5 is positive.
It fits!
5 > 0 → less than 90°
Let's look back.
First: multiply the matching parts.
Last: check.
Answer
Careful: not ⟨−3, 8⟩ ✗