Step by step
The question
Question: find the dot product of u = ⟨−3, −1⟩ and v = ⟨2, −5⟩.
u = ⟨−3, −1⟩ and v = ⟨2, −5⟩
1 Multiply the matching parts
The dot product: across times across, plus up times up.
u • v = −3 × 2 + −1 × (−5)
−3 times 2 is −6.
−1 times (−5) is 5.
u • v = −6 + 5
−6 plus 5 is −1.
So u • v = −1.
u • v = −1
Careful!
The dot product is not a vector.
We add the two products.
The answer is one number.
Careful: not ⟨−6, 5⟩ ✗
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°.
−1 is negative.
It fits!
−1 < 0 → more than 90°
Let's look back.
First: multiply the matching parts.
Last: check.
Answer
Careful: not ⟨−6, 5⟩ ✗