Vector Projection
Project one vector orthogonally onto another in 2D/3D space. Free vector projection calculator with component and scalar results for math and physics.
About This Calculator
The Vector Projection Calculator computes the orthogonal projection of one vector onto another in 2D or 3D space. Whether you are a student learning linear algebra, a physicist analyzing forces, or an engineer working with vector mechanics, this tool helps you quickly find how much of one vector points in the direction of another. Vector projection is a fundamental concept in mathematics with applications ranging from computer graphics and machine learning to structural analysis and navigation systems.
The calculator uses the standard orthogonal projection formula: proj = ((u·v) / (v·v)) x v, where u·v is the dot product of vectors u and v, and v·v is the squared magnitude (norm-squared) of vector v. The result is the vector projection of u onto v, which lies along the direction of v. The scalar projection (also called the projection length or component) is given by the signed magnitude (u·v) / |v|, representing how far u extends in the direction of v. A positive scalar projection indicates u points in the same general direction as v, while a negative value indicates the opposite direction.
Vector projection is mathematically equivalent to decomposing vector u into two perpendicular components: one parallel to v (the projection) and one orthogonal to v (the rejection). This decomposition is central to the Gram-Schmidt orthogonalization process, which is used to construct orthonormal bases in linear algebra. The orthogonal rejection vector can be found by subtracting the projection from the original vector: rej = u - proj.
In physics, vector projection is used to resolve forces into components along specific directions. For example, on an inclined plane, the gravitational force acting on an object can be decomposed into a component parallel to the slope (which causes acceleration) and a component perpendicular to the slope (which determines the normal force). In data science, vector projection forms the mathematical foundation of principal component analysis (PCA), where high-dimensional data is projected onto lower-dimensional subspaces to reduce dimensionality while preserving maximum variance.
The calculator supports both 2D and 3D vectors. For 2D vectors, simply leave the z components empty. The optional z fields default to zero, allowing seamless switching between 2D and 3D calculations. All results are displayed with four decimal places for precision, and the projection vector components are shown alongside the scalar projection value.
Frequently Asked Questions
What is the vector projection formula?
The vector projection of vector u onto vector v is given by proj = ((u·v)/(v·v)) x v, where u·v is the dot product and v·v is the squared magnitude of v. The result is a vector that lies along the direction of v and represents the orthogonal shadow of u onto v.
What is the difference between vector projection and scalar projection?
The vector projection is a vector that lies along the direction of the target vector, showing both magnitude and direction. The scalar projection (also called the projection length) is the signed magnitude of the vector projection, given by (u·v)/|v|. The scalar projection can be positive or negative depending on whether the angle between vectors is acute or obtuse.
Can I project a 2D vector onto a 3D vector?
Yes. For 2D vectors, simply leave the z components of both vectors empty (they default to zero). The calculator will treat them as 2D vectors in the xy-plane and compute the projection accordingly. The formula works for vectors of any dimension, and the calculator handles both 2D and 3D cases seamlessly.
What happens when vector v is the zero vector?
If vector v is the zero vector (all components are zero), the projection is undefined because division by zero occurs in the formula proj = ((u·v)/(v·v)) x v. The calculator gracefully returns without a result in this case, preventing any division-by-zero errors.
What are real-world applications of vector projection?
Vector projection is used extensively in physics to decompose forces (e.g., finding the component of gravity along a slope), in computer graphics for shadow calculations and lighting, in data science for principal component analysis (PCA) and linear regression, and in engineering for stress analysis and structural mechanics.
How is vector projection related to the dot product?
The dot product is the core operation in vector projection. The scalar projection (u·v)/|v| represents how much of u points in the direction of v. The vector projection multiplies this scalar by the unit vector in the direction of v, giving (u·v)/(v·v) x v. The dot product determines both the magnitude and sign of the projection.
Can the projection be longer than the original vector?
No, the scalar projection (length of the projection) cannot exceed the length of the original vector u. By the Cauchy-Schwarz inequality, |u·v| <= |u||v|, so the absolute scalar projection |u·v|/|v| <= |u|. The projection is always a component of the original vector along the target direction.