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