Step by step
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⟩ ✗