Vector Projection Calculator — Projection, Rejection & Angle

About the Vector Projection Calculator — Projection, Rejection & Angle

Vector projection decomposes a vector a into two orthogonal components: one parallel to a reference vector b (the projection) and one perpendicular to b (the rejection). This decomposition is one of the most widely used operations in linear algebra, physics, and engineering — from resolving forces along inclined planes to computing the closest point on a line.

This calculator computes both the scalar projection (comp_b(a) = a · b / ‖b‖, a signed scalar measuring how far a extends in b's direction) and the vector projection (proj_b(a) = (a · b / b · b) b, the actual vector component along b). The rejection a − proj_b(a) is the leftover perpendicular part. Together, projection + rejection perfectly reconstruct the original vector a.

The output also includes the angle between a and b, the dot product, and a relationship check (parallel, perpendicular, or general). A verification card confirms that projection and rejection are orthogonal by checking their dot product equals zero — a built-in sanity check.

The decomposition table shows a, b, projection, rejection, and their sum (which should match a) side by side. A stacked bar visualization illustrates how the magnitudes of projection and rejection partition ‖a‖, while component-wise bars break the decomposition down by axis. Six presets cover textbook scenarios including perpendicular vectors (projection = 0), parallel vectors (rejection = 0), and general cases.

Understanding vector projection is essential for least-squares regression, the Gram-Schmidt process, work computation in physics (W = F · d), and graphics (shadow casting, light reflection). This calculator makes the geometry of projection tangible through numbers and visuals.

When This Page Helps

Computing the projection requires a dot product, a second dot product (or magnitude squared), a scalar division, and a vector scaling — four operations where mixing up a and b or miscomputing the denominator yields the wrong result entirely. This calculator computes scalar projection, vector projection, rejection, and the angle between a and b for 2D and 3D vectors. It verifies orthogonality (proj · rej ≈ 0), shows a decomposition bar chart, and includes 6 presets covering textbook, perpendicular, and parallel cases. Essential for understanding work, least squares, and Gram-Schmidt.

How to Use the Inputs

1. Select 2D or 3D mode
2. Enter the components of vector a (the vector to project)
3. Enter the components of vector b (the direction to project onto)
4. Read scalar projection, vector projection, and rejection from the output cards
5. Check the decomposition table and stacked bar for visual understanding
6. Verify that proj · rej ≈ 0 (orthogonality check)

Example Calculation

Tips & Best Practices

• If a and b are perpendicular, the projection is the zero vector and the rejection equals a
• If a and b are parallel, the rejection is zero and the projection equals a (possibly scaled)
• The scalar projection is negative when the angle between a and b exceeds 90° — the shadow falls "behind" b
• Work in physics equals W = F · d = ‖F‖ · comp_d(F) — the scalar projection of force along displacement
• Swapping a and b gives a completely different projection — projection is NOT symmetric

Scalar vs. Vector Projection

The **scalar projection** of a onto b is comp_b(a) = a · b / ‖b‖, a signed number representing how far a extends in b’s direction. The **vector projection** is proj_b(a) = (a · b / b · b) b, which is the actual vector component of a along b. The projection is positive when the angle is acute, zero when perpendicular, and negative (pointing opposite to b) when obtuse. Crucially, proj_b(a) ≠ proj_a(b) — projection is **not symmetric**, and swapping a and b produces a vector in a different direction.

Rejection and Orthogonal Decomposition

The **rejection** rej_b(a) = a − proj_b(a) is the component of a perpendicular to b. Together, projection and rejection produce an **orthogonal decomposition**: a = proj_b(a) + rej_b(a), with proj ⊥ rej. This decomposition is the heart of the **Gram-Schmidt process**, which builds an orthonormal basis by repeatedly subtracting projections onto previously orthogonalized vectors. The Pythagorean theorem applies: ‖a‖² = ‖proj‖² + ‖rej‖², providing a consistency check.

Applications: Work, Least Squares, and Beyond

In **physics**, work W = F · d = ‖F‖ · comp_d(F) is the scalar projection of force along displacement. In **least squares regression**, the solution â = proj_{col(X)} b projects the observation vector onto the column space of X, minimizing the residual (rejection). In **signal processing**, projecting a signal onto a basis function extracts that frequency component (Fourier analysis). In **robotics**, projecting a desired trajectory onto the feasible space produces the closest achievable motion. Understanding projection and rejection is how mathematicians and engineers decompose complex problems into simpler, orthogonal parts.

Frequently Asked Questions

• What is the difference between scalar and vector projection?

  The scalar projection is a single number measuring how far a extends in b's direction (positive or negative). The vector projection is the actual vector component of a along b — it has both magnitude and direction (along b).

• What is vector rejection?

  The rejection is a − proj_b(a), the part of a perpendicular to b. Projection and rejection together reconstruct a: a = proj_b(a) + rej_b(a). The rejection is always orthogonal to b.

• When is the projection zero?

  The projection is zero when a is perpendicular to b (a · b = 0). In this case, a has no component along b's direction, and the rejection equals a itself.

• Is projection onto b the same as projection onto 2b?

  Yes! Scaling b doesn't change the projection because the formula proj_b(a) = (a · b / b · b) b is invariant to scaling. The direction matters, not the length.

• How does projection relate to the Gram-Schmidt process?

  Gram-Schmidt orthogonalizes vectors by iteratively subtracting projections. To make v₂ orthogonal to u₁, compute v₂ − proj_{u₁}(v₂) — this is exactly the rejection of v₂ with respect to u₁.

• What does a negative scalar projection mean?

  A negative scalar projection means the angle between a and b is obtuse (> 90°). The vector projection then points in the opposite direction to b, meaning a's component along b opposes b's direction.