Orthogonal Projection Calculator – Vector Projection
Calculate the projection of one vector onto another
Table of Contents
How to Use
- Enter the x, y, and z components of vector u (the vector to project)
- Enter the x, y, and z components of vector v (the vector to project onto)
- Click calculate to see the projection result
- View the projection vector, magnitude, and scalar projection
What is Orthogonal Projection?
The orthogonal projection of vector u onto vector v is the component of u that lies in the direction of v. Geometrically, it's the shadow that u would cast onto v if light were shining perpendicular to v.
The projection formula is: proj_v(u) = (u · v / |v|²) × v, where u · v is the dot product and |v| is the magnitude of v.
Vector vs Scalar Projection
- Vector Projection: The actual vector component of u in the direction of v. It has both magnitude and direction.
- Scalar Projection: The signed length of the projection. Positive if u and v point in similar directions, negative if opposite.
The scalar projection is calculated as: comp_v(u) = u · v / |v|
Applications of Vector Projection
- Physics: Decomposing forces into components, work calculations
- Computer Graphics: Shadow calculations, lighting models
- Machine Learning: Feature extraction, dimensionality reduction
- Engineering: Structural analysis, signal processing
- Navigation: Finding distance along a path or direction
Properties of Orthogonal Projection
- Idempotent: Projecting a projection gives the same result
- Linear: proj(au + bw) = a·proj(u) + b·proj(w)
- Orthogonal Complement: u - proj_v(u) is perpendicular to v
- Magnitude Bound: |proj_v(u)| ≤ |u|
Frequently Asked Questions
- What is the difference between vector and scalar projection?
- Vector projection gives you a vector in the direction of v with the appropriate length. Scalar projection gives you just the signed length (a number) of that projection. The vector projection equals the scalar projection times the unit vector in the direction of v.
- What happens if I project onto a zero vector?
- Projecting onto a zero vector is undefined mathematically because division by zero occurs. Our calculator returns a zero vector in this case to handle the edge case gracefully.
- How is projection related to the dot product?
- The dot product u · v equals |u| |v| cos(θ), where θ is the angle between vectors. The scalar projection is u · v / |v| = |u| cos(θ), which is the component of u in the direction of v.
- Can projection be negative?
- The scalar projection can be negative when the angle between vectors is greater than 90°. The vector projection points in the opposite direction of v in this case.