Vector Projection Calculator
Calculate vector projections with comprehensive step-by-step solutions and geometric analysis. Our linear algebra calculator supports 2D and 3D vectors, scalar projections, dot products, and detailed mathematical explanations for advanced mathematics.
Last updated: December 15, 2024
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Projection Formula:
projb(a) = ((a·b) / |b|²) × b
Projection Analysis
Vector Projection:
⟨3.0000, 0.0000⟩
Analysis Type:
Step-by-step solution
Scalar Projection:
3.0000
Projection Magnitude:
3.0000
Dot Product (a·b):
3.0000
Angle:
53.13°
Vector Magnitudes:
Solution Steps:
- 1. Given vectors: a = ⟨3.0000, 4.0000⟩, b = ⟨1.0000, 0.0000⟩
- 2. Calculate dot product: a·b = 3(1) + 4(0) = 3.0000
- 3. Calculate |b|²: 1² + 0² = 1.0000
- 4. Calculate |b|: √1.0000 = 1.0000
- 5. Scalar projection: comp_b(a) = (a·b)/|b| = 3.0000/1.0000 = 3.0000
- 6. Vector projection: proj_b(a) = ((a·b)/|b|²) × b = (3.0000/1.0000) × ⟨1.0000, 0.0000⟩
- 7. proj_b(a) = ⟨3.0000, 0.0000⟩
- 8. |proj_b(a)| = 3.0000
Analysis:
The projection of vector a onto vector b is ⟨3.0000, 0.0000⟩ with magnitude 3.0000. The scalar projection is 3.0000.
Vector Projection Tips:
- • Formula: projb(a) = ((a·b) / |b|²) × b
- • Scalar projection: compb(a) = (a·b) / |b|
- • Geometric meaning: "Shadow" of a onto b
- • Direction: Same as b if a·b > 0, opposite if a·b < 0
Quick Example Result
For projecting vector ⟨3, 4⟩ onto ⟨1, 0⟩:
proj = ⟨3.0000, 0.0000⟩
Scalar projection: 3.0000, Angle: 53.13°
How This Calculator Works
Our vector projection calculator applies fundamental linear algebra principles to compute the projection of one vector onto another. The calculator uses dot product operations and vector magnitude calculations to determine both scalar and vector projections, providing geometric insights and mathematical analysis for 2D and 3D vector spaces.
Vector Projection Algorithm
a·b = a₁b₁ + a₂b₂ + a₃b₃|a| = √(a₁² + a₂² + a₃²), |b| = √(b₁² + b₂² + b₃²)comp_b(a) = (a·b) / |b|proj_b(a) = ((a·b) / |b|²) × bThe vector projection algorithm systematically applies linear algebra operations to decompose vectors. The scalar projection comp_b(a) = (a·b)/|b| gives the signed length of the projection, while the vector projection proj_b(a) = ((a·b)/|b|²)b provides both magnitude and direction. The geometric interpretation represents the "shadow" of vector a when projected onto the line containing vector b.
Mathematical foundation for vector projection operations
Mathematical Foundation
Vector projection is fundamentally based on the dot product and the geometric relationship between vectors in Euclidean space. The projection of vector a onto vector b represents the component of a that lies in the direction of b. This operation is essential in linear algebra, as it allows us to decompose vectors into parallel and perpendicular components. The scalar projection gives us the signed magnitude, while the vector projection provides the actual vector result. The formula proj_b(a) = ((a·b)/|b|²)b ensures that the result lies along the direction of b with the correct magnitude.
- Dot product measures the similarity and alignment between vectors
- Projection decomposes vectors into parallel and perpendicular components
- Scalar projection provides signed magnitude information
- Vector projection gives both direction and magnitude of the component
Sources & References
- Linear Algebra and Its Applications - David C. Lay, Steven R. Lay, Judi J. McDonaldComprehensive treatment of vector operations and projections
- Wolfram MathWorld - Vector Projection ReferenceDetailed mathematical reference with geometric interpretations
- Introduction to Linear Algebra - Gilbert StrangEducational treatment of vector spaces and linear transformations
Need help with other vector operations? Check out our dot product calculator and cross product calculator.
Get Custom Calculator for Your PlatformExample Analysis
Problem Setup:
- Applied Force: F = ⟨50, 30⟩ N
- Direction: d = ⟨3, 4⟩
- Goal: Find force component along direction d
- Application: Engineering stress analysis
Solution Process:
Result: Force component = ⟨32.4, 43.2⟩ N, Magnitude = 54.0 N
This engineering example demonstrates how vector projection helps analyze forces in structural design. The applied force F = ⟨50, 30⟩ N has a component of 54.0 N along the direction ⟨3, 4⟩. The calculation shows: F·d = 50×3 + 30×4 = 270, |d| = √(9+16) = 5, so the scalar projection is 270/5 = 54.0 N. The vector projection is (270/25) × ⟨3, 4⟩ = ⟨32.4, 43.2⟩ N. This represents the effective force component that acts along the specified direction, which is crucial for determining stress, strain, and structural integrity in engineering applications. The remaining component F - proj_d(F) = ⟨17.6, -13.2⟩ N acts perpendicular to direction d, contributing to shear forces in the structure.
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