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Vector Projection

Projection of A onto B: proj_B(A) = (A·B/|B|²)B. Scalar projection = A·B̂ = A·B/|B|. Rejection = A − proj_B(A) is perpendicular to B. A = proj + rej.

Concept Fundamentals
comp_B A = A·B/|B|
Scalar
proj_B A = (A·B/|B|²)B
Vector
A − proj_B A
Rejection
A = proj + rej
Decompose

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proj_B(A) + rej_B(A) = A. rej_B(A) · B = 0. When B is unit vector: proj = (A·B)B.

Key quantities
comp_B A = A·B/|B|
Scalar
Key relation
proj_B A = (A·B/|B|²)B
Vector
Key relation
A − proj_B A
Rejection
Key relation
A = proj + rej
Decompose
Key relation

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Why: Projections decompose vectors into parallel and perpendicular components. Used in physics (work), computer graphics (shadows), and regression (least squares).

How: Scalar projection = A·B/|B| (signed length). Vector projection = (A·B/|B|²)B. Rejection = A − proj_B(A). Rejection is perpendicular to B.

proj_B(A) + rej_B(A) = A.rej_B(A) · B = 0.
Sources:Khan Academy — Vector ProjectionWolfram MathWorld — Projection

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Compute ProjectionEnter vectors A and B

Vectors A and B

Vector A (to project)

Vector B (onto)

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

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proj_B(A) = (A·B/|B|²)B.

— Vector Projection

⊥

Rejection is perpendicular to B.

— Decomposition

Key Takeaways

  • • Scalar projection: comp_B(A) = (A·B)/|B| — the signed length of A along B.
  • • Vector projection: proj_B(A) = ((A·B)/|B|²)B — the component of A parallel to B.
  • • Rejection: rej_B(A) = A - proj_B(A) — the component of A perpendicular to B.
  • • A = proj_B(A) + rej_B(A) — every vector decomposes into parallel and perpendicular parts.
  • • If A ⊥ B, then proj_B(A)=0 and rej_B(A)=A. If A ∥ B, then rej_B(A)=0.

Did You Know?

Work in Physics

Work W = F·d is the scalar projection of force onto displacement. Only the parallel component does work.

Shadow

The projection of a vector onto another is like the "shadow" one vector casts onto the other.

Orthogonal Decomposition

proj and rej are orthogonal: proj_B(A)·rej_B(A)=0. They form a right triangle with A.

Gram-Schmidt

The Gram-Schmidt process uses projections to build orthogonal bases from a set of vectors.

Least Squares

Linear regression finds the projection of the data vector onto the column space of the design matrix.

Computer Graphics

Projections are used for shadows, camera views, and resolving forces in game physics.

Understanding Projection

The projection of A⃗\vec{A} onto B⃗\vec{B} gives the component of A in the direction of B:

projB⃗A⃗=A⃗⋅B⃗∣B⃗∣2B⃗,compB⃗A⃗=A⃗⋅B⃗∣B⃗∣\text{proj}_{\vec{B}}\vec{A} = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|^2}\vec{B}, \quad \text{comp}_{\vec{B}}\vec{A} = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|}

Expert Tips

Zero B

Cannot project onto the zero vector. |B| must be non-zero.

Signed Scalar

Scalar projection can be negative if the angle between A and B is obtuse.

Unit B

If B is a unit vector, proj_B(A) = (A·B)B and comp_B(A) = A·B.

Verification

Check that proj + rej = A and proj·rej = 0.

FAQ

What is scalar projection?

comp_B(A) = (A·B)/|B|. It is the signed length of the projection — positive if A and B form an acute angle.

What is vector projection?

proj_B(A) = ((A·B)/|B|²)B. It is the vector component of A parallel to B.

What is the rejection?

rej_B(A) = A - proj_B(A). It is the component of A perpendicular to B.

When is the scalar projection negative?

When the angle between A and B is greater than 90°. The projection points opposite to B.

When is proj_B(A) = A?

When A is parallel to B. Then rej_B(A) = 0.

When is proj_B(A) = 0?

When A is perpendicular to B (A·B=0). Then rej_B(A) = A.

How is this used in physics?

To resolve forces into components, compute work (F·d), and analyze motion on inclined planes.

How to Use

  1. Enter components of vector A (to be projected) and vector B (onto which to project).
  2. Click a sample example or Calculate.
  3. Review scalar projection, vector projection, rejection, and visualization.
  4. Copy results if needed.

Disclaimer: Cannot project onto the zero vector. Use z=0 for 2D vectors.

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