Step by step
The question
Question: find the dot product of u = ⟨3, −1, 2⟩ and v = ⟨1, 2, −4⟩.
⟨3, −1, 2⟩ • ⟨1, 2, −4⟩
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 = 3 × 1 + −1 × 2 + 2 × (−4)
3 times 1 is 3.
−1 times 2 is −2.
2 times (−4) is −8.
u • v = 3 − 2 − 8
Add them up: u • v = −7.
u • v = −7
Careful!
The dot product is not a vector.
We add the three products.
The answer is one number.
Careful: not ⟨3, −2, −8⟩ ✗
2 Check
Let's check: the other order, v dot u.
We get −7 again.
Right!
v • u = 1 × 3 + 2 × (−1) + −4 × 2 = −7
A negative dot product means the arrows point in opposite ways: the angle is more than 90°.
−7 < 0 → angle more than 90°
Let's look back.
First: multiply the matching parts.
Last: check.
Answer
Careful: not ⟨3, −2, −8⟩ ✗