Orthogonality
Vectors are orthogonal when their dot product is 0.
Key idea
Vectors are orthogonal when their dot product is 0.
In depth
Vectors are orthogonal (perpendicular) exactly when a · b = 0. This generalises right angles to any dimension and is the basis of coordinates, projections, and least squares.
Worked example
(2, 3) · (−3, 2) = −6 + 6 = 0, so they are orthogonal.
Formula
Real-world uses
- Checking that two forces act independently
- Choosing independent features in data analysis
Common mistake
Assuming the zero dot product means one vector is zero. It only means they are perpendicular (or one is zero).
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 Orthogonality?
Vectors are orthogonal when their dot product is 0. It is a topic at the advanced level in Linear Algebra.
What is the formula for Orthogonality?
The key formula is u·v = 0.
Where is Orthogonality used in real life?
Real-world examples: Checking that two forces act independently; Choosing independent features in data analysis.
What should I learn before Orthogonality?
Make sure you are comfortable with Angle Between Vectors, the previous topic in Linear Algebra.
What is a common mistake with Orthogonality?
Assuming the zero dot product means one vector is zero. It only means they are perpendicular (or one is zero).
How can I practise Orthogonality?
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 Orthogonality?
Move on to Projection next.
Related advanced topics in Linear Algebra
- Dot Product – Sum of component products.
- Angle Between Vectors – Cosine from the dot product.
- 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.