Lesson 8 of 8
Dot product basics
Multiply matching components and add: two vectors in, one number out.
The idea
The dot product combines two vectors into a single number. Multiply the matching components, then add the products. It is written with a raised dot: u · v.
u equals the vector with components 2, 3 v equals the vector with components 4, 1
u dot v equals 2 times 4 plus 3 times 1 equals 8 plus 3 equals 11
The answer is a scalar
Adding and subtracting vectors gives you another vector. The dot product is different: two vectors go in and one ordinary number comes out. That is why the answer box for a dot product is a single box.
With negative numbers
Work out each product carefully first, including its sign, and only then add.
the vector with components 1, negative 2 dot the vector with components 3, 4 equals 1 times 3 plus negative 2 times 4 equals 3 plus negative 8 equals negative 5
Worked example
u equals the vector with components 5, 2 v equals the vector with components 3, 4
Find the dot product udotv.
Multiply matching components
First components: 5times3equals15
Second components: 2times4equals8
Add the products
u dot v equals 15 plus 8 equals 23Notice the type of answer
The dot product of two vectors is a scalar — a single number, not a vector.
Common mistakes
- Stopping after one product. There are two products; add them both.
- Pairing across. Multiply first with first and second with second, never first with second.
- Writing a vector. The dot product is one number.