Step by step
The question
Question: find the dot product of u = ⟨2, 3⟩ and v = ⟨−3, 2⟩.
u = ⟨2, 3⟩ and v = ⟨−3, 2⟩
1 Multiply the matching parts
The dot product: across times across, plus up times up.
u • v = 2 × (−3) + 3 × 2
2 times (−3) is −6.
3 times 2 is 6.
u • v = −6 + 6
−6 plus 6 is 0.
So u • v = 0.
u • v = 0
Careful!
The dot product is not a vector.
We add the two products.
The answer is one number.
Careful: not ⟨−6, 6⟩ ✗
2 Check
Does it make sense?
Let's draw u and v.
They meet at a square corner.
A dot product of 0 means a square corner: the vectors are perpendicular.
It fits!
0 → exactly 90°
Let's look back.
First: multiply the matching parts.
Last: check.
Answer
Careful: not ⟨−6, 6⟩ ✗