Step by step
The question
Question: find the dot product of u = ⟨1, 1, 1⟩ and v = ⟨2, 3, 4⟩.
⟨1, 1, 1⟩ • ⟨2, 3, 4⟩
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 × 2 + 1 × 3 + 1 × 4
1 times 2 is 2.
1 times 3 is 3.
1 times 4 is 4.
u • v = 2 + 3 + 4
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 ⟨2, 3, 4⟩ ✗
2 Check
Let's check: the other order, v dot u.
We get 9 again.
Right!
v • u = 2 × 1 + 3 × 1 + 4 × 1 = 9
A positive dot product means the arrows point in a similar way: the angle is less than 90°.
9 > 0 → angle less than 90°
Let's look back.
First: multiply the matching parts.
Last: check.
Answer
Careful: not ⟨2, 3, 4⟩ ✗