📐 Vector Projection Calculator
Project one vector onto another
Vector A (to project)
Vector B (to project onto)
How to Use This Calculator
Enter Vector A Components
Input the x, y, and optionally z components of vector A (the vector to project).
Enter Vector B Components
Input the x, y, and optionally z components of vector B (the vector to project onto). B cannot be zero vector.
Calculate
Press "Calculate Projection" to find proj_B(A) = ((A·B)/(B·B))B.
Formula
proj_B(A) = ((A·B)/(B·B)) × B
where A·B is the dot product and B·B = |B|²
Example: A = (3, 4), B = (1, 0)
A·B = 3×1 + 4×0 = 3
B·B = 1² + 0² = 1
proj_B(A) = (3/1) × (1, 0) = (3, 0)
Magnitude: |proj_B(A)| = 3
About Vector Projection Calculator
The Vector Projection Calculator finds the projection of one vector onto another. The projection is the component of A in the direction of B, calculated as proj_B(A) = ((A·B)/(B·B)) × B.
When to Use This Calculator
- Physics: Decompose forces into components along a direction
- Engineering: Resolve forces and displacements along axes
- Mathematics: Perform vector decompositions
- Computer Graphics: Calculate projections for rendering
- Signal Processing: Project signals onto subspaces
Understanding Vector Projection
Vector projection finds the component of A in the direction of B. The projection is parallel to B and has magnitude |A|cos(θ) where θ is the angle between A and B. The perpendicular component is A - proj_B(A).
Frequently Asked Questions
What is vector projection?
Vector projection is the component of vector A in the direction of vector B. Formula: proj_B(A) = ((A·B)/(B·B)) × B. It's the closest point to A that's parallel to B.
What does the projection magnitude represent?
The magnitude |proj_B(A)| = |A|cos(θ) represents the length of the projection, where θ is the angle between A and B. It's the component of A along B's direction.
What is the perpendicular component?
The perpendicular component is A - proj_B(A). It's the part of A that's orthogonal (perpendicular) to B. Together, proj_B(A) and (A - proj_B(A)) sum to A.
Can I project onto a zero vector?
No! Projection onto a zero vector is undefined (division by zero). Vector B must be non-zero (|B| > 0) for projection to exist.