Dot Product
Sum of component products.
Key idea
Sum of component products.
In depth
a · b = a₁b₁ + a₂b₂ + ... = |a||b| cos θ. It measures how aligned two vectors are: positive for similar directions, zero for perpendicular, negative for opposite. The result is a scalar.
Worked example
(1, 2, 3) · (4, −1, 2) = 4 − 2 + 6 = 8.
Formula
Real-world uses
- Calculating the work done by a force
- Scoring similarity between search queries and documents
Common mistake
Writing a vector as the answer.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Dot Product?
Sum of component products. It is a topic at the advanced level in Linear Algebra.
What is the formula for Dot Product?
The key formula is u·v = Σuᵢvᵢ.
Where is Dot Product used in real life?
Real-world examples: Calculating the work done by a force; Scoring similarity between search queries and documents.
What should I learn before Dot Product?
Make sure you are comfortable with Unit Vectors, the previous topic in Linear Algebra.
What is a common mistake with Dot Product?
Writing a vector as the answer.
How can I practise Dot Product?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Dot Product?
Move on to Angle Between Vectors next.
Related advanced topics in Linear Algebra
- Angle Between Vectors – Cosine from the dot product.
- Orthogonality – Vectors are orthogonal when their dot product is 0.
- Projection – Component of one vector along another.
- Cross Product – A vector perpendicular to two vectors in 3D.
- Lines and Planes in Vector Form – Parametric and normal equations.
- Matrix Multiplication – Row-by-column products; not commutative.