Vector Projection Calculator
Calculate the projection of one vector onto another with detailed step-by-step solutions. Visualize vector projections, understand orthogonal components, and explore the geometric interpretation of vector projections in 2D and 3D space.
Vector Input
Vector Projection Theory
What is Vector Projection?
Vector projection is the operation of projecting one vector onto another vector. The projection of vector v onto vector u is the component of v that lies in the direction of u.
Vector Projection Formula:
proju(v) = ((v · u) / (u · u)) × u
Scalar Projection Formula:
compu(v) = (v · u) / ||u||
Where: v · u is the dot product, ||u|| is the magnitude of u
Calculation Steps
Step 1: Calculate Dot Product
Compute v · u = v₁u₁ + v₂u₂ + ... + vₙuₙ
Step 2: Calculate u · u
Compute u · u = u₁² + u₂² + ... + uₙ²
Step 3: Find Scalar Factor
Calculate k = (v · u) / (u · u)
Step 4: Multiply by Direction
proj_u(v) = k × u = k(u₁, u₂, ..., uₙ)
Applications & Properties
Physics
Force components, work calculation, motion analysis
Computer Graphics
Lighting calculations, shadow mapping, collision detection
Engineering
Structural analysis, signal processing, optimization
Machine Learning
Principal component analysis, feature extraction
Geometry
Distance calculations, orthogonal decomposition
Statistics
Regression analysis, least squares fitting
Worked Examples
Example 1: 2D Vector Projection
Problem:
Find the projection of v = [3, 4] onto u = [1, 2]
Solution:
Step 1: v · u = (3)(1) + (4)(2) = 3 + 8 = 11
Step 2: u · u = (1)² + (2)² = 1 + 4 = 5
Step 3: k = 11/5 = 2.2
Step 4: proj_u(v) = 2.2 × [1, 2] = [2.2, 4.4]
Result: The projection vector is [2.2, 4.4]
Example 2: 3D Vector Projection
Problem:
Find the projection of v = [2, 1, 3] onto u = [1, 0, 1]
Solution:
Step 1: v · u = (2)(1) + (1)(0) + (3)(1) = 2 + 0 + 3 = 5
Step 2: u · u = (1)² + (0)² + (1)² = 1 + 0 + 1 = 2
Step 3: k = 5/2 = 2.5
Step 4: proj_u(v) = 2.5 × [1, 0, 1] = [2.5, 0, 2.5]
Result: The projection vector is [2.5, 0, 2.5]
Frequently Asked Questions
Vector projection gives you a vector in the direction of u, while scalar projection gives you just the length (magnitude) of that projection. Scalar projection can be negative if the angle between vectors is obtuse.
Projection onto a zero vector is undefined because we would be dividing by zero in the formula. The zero vector has no direction, so projection onto it doesn't make mathematical sense.
The dot product v·u equals ||v|| × ||u|| × cos(θ), where θ is the angle between vectors. The scalar projection is v·u / ||u||, which equals ||v|| × cos(θ) - the "shadow" of v on u.
No, the magnitude of the projection is always less than or equal to the magnitude of the original vector. It equals the original magnitude only when the vectors are parallel (angle = 0°).
Any vector v can be decomposed into two orthogonal components: one parallel to u (the projection) and one perpendicular to u. These components are perpendicular to each other and sum to the original vector.
Check that: (1) The projection is parallel to u (one is a scalar multiple of the other), (2) The perpendicular component is orthogonal to u (their dot product is zero), (3) Projection + perpendicular = original vector.