Step by step
The question
Question: find the dot product of u = ⟨1, 2, 3⟩ and v = ⟨4, −5, 6⟩.
⟨1, 2, 3⟩ • ⟨4, −5, 6⟩
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 = 1 × 4 + 2 × (−5) + 3 × 6
1 times 4 is 4.
2 times (−5) is −10.
3 times 6 is 18.
u • v = 4 − 10 + 18
Add them up: u • v = 12.
u • v = 12
Careful!
The dot product is not a vector.
We add the three products.
The answer is one number.
Careful: not ⟨4, −10, 18⟩ ✗
2 Check
Let's check: the other order, v dot u.
We get 12 again.
Right!
v • u = 4 × 1 + −5 × 2 + 6 × 3 = 12
A positive dot product means the arrows point in a similar way: the angle is less than 90°.
12 > 0 → angle less than 90°
Let's look back.
First: multiply the matching parts.
Last: check.
Answer
Careful: not ⟨4, −10, 18⟩ ✗