Step by step
The question
Question: find the dot product of u = ⟨−2, 5, 3⟩ and v = ⟨4, 1, −2⟩.
⟨−2, 5, 3⟩ • ⟨4, 1, −2⟩
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 × 4 + 5 × 1 + 3 × (−2)
−2 times 4 is −8.
5 times 1 is 5.
3 times (−2) is −6.
u • v = −8 + 5 − 6
Add them up: u • v = −9.
u • v = −9
Careful!
The dot product is not a vector.
We add the three products.
The answer is one number.
Careful: not ⟨−8, 5, −6⟩ ✗
2 Check
Let's check: the other order, v dot u.
We get −9 again.
Right!
v • u = 4 × (−2) + 1 × 5 + −2 × 3 = −9
A negative dot product means the arrows point in opposite ways: the angle is more than 90°.
−9 < 0 → angle more than 90°
Let's look back.
First: multiply the matching parts.
Last: check.
Answer
Careful: not ⟨−8, 5, −6⟩ ✗