Step by step
The question
Question: find the dot product of u = ⟨1, 2, −1⟩ and v = ⟨3, −1, 1⟩.
⟨1, 2, −1⟩ • ⟨3, −1, 1⟩
1 Multiply the matching parts
The dot product: multiply the matching parts: x times x, y times y, z times z.
Then add.
u • v = 1 × 3 + 2 × (−1) + −1 × 1
1 times 3 is 3.
2 times (−1) is −2.
−1 times 1 is −1.
u • v = 3 − 2 − 1
Add them up: u • v = 0.
u • v = 0
Careful!
The dot product is not a vector.
We add the three products.
The answer is one number.
Careful: not ⟨3, −2, −1⟩ ✗
2 Check
Let's check: the other order, v dot u.
We get 0 again.
Right!
v • u = 3 × 1 + −1 × 2 + 1 × (−1) = 0
A dot product of 0 means a square corner: the vectors are perpendicular.
0 → exactly 90°
Let's look back.
First: multiply the matching parts.
Last: check.
Answer
Careful: not ⟨3, −2, −1⟩ ✗