Vector Projection Calculator
Enter two 3D vectors to compute the projection of a onto b. a=(3,4,0), b=(1,0,0) → scalar projection=3, vector projection=(3,0,0).
Input Data
Results
At a glance:Vector projection proj_b(a) is the component of a along the direction of b. Scalar projection (component length) = a·b/|b| = |a|cosθ (θ is the angle between them). Vector projection = (a·b/|b|²)·b. Properties: when a is parallel to b the projection equals a; perpendicular gives 0; anti-parallel gives a negative scalar. Physical meaning: decompose a force into its effective component (along an incline), motion components, signal projection. Applications: (1) force decomposition (incline, rope tension); (2) velocity components (horizontal/vertical); (3) least-squares regression; (4) projection geometry (shadows, views); (5) signal processing (orthogonal decomposition).
Formula
Scalar projection: comp_b(a) = a·b / |b| = |a|cosθ
Vector projection: proj_b(a) = (a·b / |b|²) · b
Dot product: a·b = ax·bx + ay·by + az·bz
Perpendicular component: a - proj_b(a)
$$\text{proj}_{\mathbf{b}}(\mathbf{a}) = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|^2}\mathbf{b}, \quad \text{comp}_{\mathbf{b}}(\mathbf{a}) = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|} = |\mathbf{a}|\cos\theta$$How to Use
- Enter the three components of vector a (ax, ay, az).
- Enter the three components of vector b (bx, by, bz).
- The tool computes the scalar projection and vector projection components.
Case Studies
Force Decomposition and Inclines
Incline: gravity F=(0,-mg,0), incline direction b=(cosθ, -sinθ,0). Scalar projection = mg·sinθ is the component along the incline. Steeper incline → larger sliding force.
Rope tension: the projection of force F onto the rope direction = effective pull. With multiple ropes, each projection component determines equilibrium.
Wind projection: wind velocity vector projected onto a building's normal = wind pressure component. Structural design must compute the effective wind-facing area.
Motion Components and Least Squares
Velocity component: v=(3,4,0) projected onto the x direction = 3. Horizontal motion 3 m/s, vertical 4 m/s. Projectile motion decomposition.
Least squares: the y vector projected onto the column space of X = best-fit ŷ. Regression coefficients = projection ratios.
Signal projection: signal projected onto an orthogonal basis = spectral component. The FFT is essentially projection onto a sine basis.
FAQ
What is the difference between scalar and vector projection?
Scalar projection comp_b(a)=a·b/|b|=|a|cosθ is a number (can be negative). Vector projection proj_b(a)=(a·b/|b|²)·b is a vector (along b). The scalar projection is the magnitude (signed) of the vector projection.
Why is the projection of perpendicular vectors zero?
At θ=90°, cosθ=0, so scalar projection=|a|cos90°=0. Vector a has no component in the b direction. Physically: a perpendicular force does no work along that direction (W=F·d cosθ=0). Perpendicular motion has no component along that direction.
What is the relationship between projection and dot product?
Scalar projection comp_b(a)=a·b/|b|. Thus a·b=|b|·comp_b(a)=scalar projection × |b|. The dot product is essentially the length of b times the component of a along b. Geometrically: a·b=|a||b|cosθ.
Can the projection be larger than the original vector?
No. |scalar projection|=|a||cosθ|≤|a| (since |cosθ|≤1). At θ=0° the projection=|a| is maximal. The magnitude of the vector projection ≤ |a|. Projection is a component, never larger than the whole.
How do I find the perpendicular component?
The component of a perpendicular to b = a - proj_b(a). It is orthogonal to b. Property: proj_b(a) + perpendicular component = a (orthogonal decomposition). Applications: decompose a force into effective and perpendicular components, find orthogonal components.
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References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.