Projection of a Vector: Scalar Formula Explained Simply

The projection of vector a onto vector b is the "shadow" of a cast along the direction of b. It tells you how much of a points in b's direction, and it splits naturally into a number, the scalar projection, and a vector, the vector projection.

Projections are everywhere in physics: the work done by a force is the force's projection along the displacement times the distance, and resolving any vector into components is just projecting it onto the coordinate axes.

Key Concepts

• Scalar projection (also called the component) of a on b: (a · b) / |b| = |a| cos θ, where θ is the angle between a and b.
• It is a signed number: positive when θ is acute, negative when θ is obtuse, zero when the vectors are perpendicular.
• Vector projection: ((a · b) / |b|²) b — the scalar projection multiplied by the unit vector in b's direction.
• The leftover part, a − proj_b(a), is perpendicular to b; it is the component of a orthogonal to b.
• You cannot project onto the zero vector, because the formulas divide by |b|.
• Units: the scalar projection has the units of a, and the vector projection is a vector with the units of a.

Formulas

• Scalar projection of a on b: (a · b) / |b|.
• Vector projection of a on b: ((a · b) / |b|²) b.
• Note the denominators differ: |b| for the scalar version, |b|² for the vector version — the most common slip.
• Orthogonal component: a − ((a · b) / |b|²) b, which satisfies (a − proj) · b = 0.

Worked Example

Project a = (3, 4, 0) onto b = (2, 2, 1).

1. Dot product: a · b = 3×2 + 4×2 + 0×1 = 6 + 8 + 0 = 14.
2. |b| = √(4 + 4 + 1) = √9 = 3.
3. Scalar projection: 14 / 3 ≈ 4.67.
4. Vector projection: (14/9)(2, 2, 1) = (28/9, 28/9, 14/9) ≈ (3.11, 3.11, 1.56).
5. Check: the length of the vector projection is (14/9) × 3 = 14/3 ≈ 4.67, matching the scalar projection. Correct.
6. Orthogonal remainder: a − proj = (3 − 28/9, 4 − 28/9, 0 − 14/9) = (−1/9, 8/9, −14/9); dotting with b gives (−2 − 16 − 14)/9… let us verify: (−1/9)×2 + (8/9)×2 + (−14/9)×1 = (−2 + 16 − 14)/9 = 0. Perpendicular, as required.

Common Misconceptions

• Projection is always positive. The scalar projection carries the sign of cos θ; an obtuse angle gives a negative projection.
• Projecting a on b equals projecting b on a. False: (a · b)/|b| versus (a · b)/|a| differ unless |a| = |b|.
• The vector projection formula has |b| in the denominator. It is |b|². Using |b| produces a vector of the wrong length.

Real-World Applications

• Physics: work W = F · d uses the force's projection along the displacement.
• Computer graphics: shadows and orthographic projections onto planes.
• Resolving forces into components along ramps and axes in mechanics.
• Signal processing: projecting a signal onto basis directions.

Projecting a onto b keeps only the part of a along b: scalar part (a · b)/|b|, vector part ((a · b)/|b|²) b. Watch the sign, use |b|² for the vector form, and remember the remainder is perpendicular to b.