Step by step
The question
Question: find the dot product of u = ⟨5, −3, 2⟩ and v = ⟨2, 4, −1⟩.
⟨5, −3, 2⟩ • ⟨2, 4, −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 = 5 × 2 + −3 × 4 + 2 × (−1)
5 times 2 is 10.
−3 times 4 is −12.
2 times (−1) is −2.
u • v = 10 − 12 − 2
Add them up: u • v = −4.
u • v = −4
Careful!
The dot product is not a vector.
We add the three products.
The answer is one number.
Careful: not ⟨10, −12, −2⟩ ✗
2 Check
Let's check: the other order, v dot u.
We get −4 again.
Right!
v • u = 2 × 5 + 4 × (−3) + −1 × 2 = −4
A negative dot product means the arrows point in opposite ways: the angle is more than 90°.
−4 < 0 → angle more than 90°
Let's look back.
First: multiply the matching parts.
Last: check.
Answer
Careful: not ⟨10, −12, −2⟩ ✗