Projection
Component of one vector along another.
Key idea
Component of one vector along another.
In depth
The projection of b onto a is proj_a(b) = ((a · b)/(a · a)) a: the shadow of b along a. The leftover b − proj is perpendicular to a. It is the closest point on the line to b.
Worked example
b = (3, 4) onto a = (1, 0): ((3)/(1))(1, 0) = (3, 0).
Formula
Real-world uses
- Finding the shadow of a force along a beam
- Removing noise from a signal in data science
Common mistake
Dividing by |a| instead of a · a when the answer is the vector projection.
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 Projection?
Component of one vector along another. It is a topic at the advanced level in Linear Algebra.
What is the formula for Projection?
The key formula is proj_u v = (v·u/u·u) u.
Where is Projection used in real life?
Real-world examples: Finding the shadow of a force along a beam; Removing noise from a signal in data science.
What should I learn before Projection?
Make sure you are comfortable with Orthogonality, the previous topic in Linear Algebra.
What is a common mistake with Projection?
Dividing by |a| instead of a · a when the answer is the vector projection.
How can I practise Projection?
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 Projection?
Move on to Cross Product next.
Related advanced topics in Linear Algebra
- Dot Product – Sum of component products.
- Angle Between Vectors – Cosine from the dot product.
- Orthogonality – Vectors are orthogonal when their dot product is 0.
- 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.