Step by step
The question
Question: find the dot product of u = ⟨2, 0, 1⟩ and v = ⟨3, 4, 5⟩.
⟨2, 0, 1⟩ • ⟨3, 4, 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 = 2 × 3 + 0 × 4 + 1 × 5
2 times 3 is 6.
0 times 4 is 0.
1 times 5 is 5.
u • v = 6 + 0 + 5
Add them up: u • v = 11.
u • v = 11
Careful!
The dot product is not a vector.
We add the three products.
The answer is one number.
Careful: not ⟨6, 0, 5⟩ ✗
2 Check
Let's check: the other order, v dot u.
We get 11 again.
Right!
v • u = 3 × 2 + 4 × 0 + 5 × 1 = 11
A positive dot product means the arrows point in a similar way: the angle is less than 90°.
11 > 0 → angle less than 90°
Let's look back.
First: multiply the matching parts.
Last: check.
Answer
Careful: not ⟨6, 0, 5⟩ ✗