Step by step
The question
Question: find the dot product of u = ⟨2, 3, −1⟩ and v = ⟨1, −1, 1⟩.
⟨2, 3, −1⟩ • ⟨1, −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 = 2 × 1 + 3 × (−1) + −1 × 1
2 times 1 is 2.
3 times (−1) is −3.
−1 times 1 is −1.
u • v = 2 − 3 − 1
Add them up: u • v = −2.
u • v = −2
Careful!
The dot product is not a vector.
We add the three products.
The answer is one number.
Careful: not ⟨2, −3, −1⟩ ✗
2 Check
Let's check: the other order, v dot u.
We get −2 again.
Right!
v • u = 1 × 2 + −1 × 3 + 1 × (−1) = −2
A negative dot product means the arrows point in opposite ways: the angle is more than 90°.
−2 < 0 → angle more than 90°
Let's look back.
First: multiply the matching parts.
Last: check.
Answer
Careful: not ⟨2, −3, −1⟩ ✗