orthogonal projection calculator

Calculate the orthogonal projection of a vector v onto a subspace spanned by one or more orthogonal vectors u1, u2, ....

Vector v (e.g., 1,2,3):

Subspace basis vectors u (one vector per line, e.g., 1,0,0
0,1,0). Assumes these vectors are orthogonal.

In the vast and intricate world of linear algebra, understanding how vectors relate to subspaces is fundamental. One of the most powerful tools for this purpose is the orthogonal projection. This calculator helps you determine the orthogonal projection of a vector onto a subspace, a concept vital in fields ranging from computer graphics to data science.

What is Orthogonal Projection?

Imagine you have a vector, like an arrow pointing in space, and a flat surface (a subspace). The orthogonal projection of that vector onto the surface is essentially its "shadow" cast directly onto the surface by a light source positioned infinitely far away and perpendicular to the surface. It's the closest point in the subspace to the original vector.

More formally, the orthogonal projection of a vector v onto a subspace W is the vector p in W such that (v - p) is orthogonal to every vector in W. This means the difference between the original vector and its projection is perpendicular to the subspace itself.

The Mathematics Behind It

Projecting onto a Single Vector

The simplest case is projecting a vector v onto another non-zero vector u. The formula for this is:

proj_u(v) = ((v · u) / ||u||²) * u

v · u represents the dot product of vectors v and u. It's a scalar value that measures how much v points in the direction of u.
||u||² represents the squared Euclidean norm (or length) of vector u. It's calculated as u · u.
The result is a scalar ((v · u) / ||u||²) multiplied by the vector u, yielding a new vector that lies along u.

Projecting onto a Subspace

When you want to project a vector v onto a subspace W, the process becomes slightly more involved. If you have an orthogonal basis for the subspace W, say {u₁, u₂, ..., u_k}, then the projection of v onto W is simply the sum of the projections of v onto each of these basis vectors:

proj_W(v) = proj_u1(v) + proj_u2(v) + ... + proj_uk(v)

It's crucial that the basis vectors u_i are mutually orthogonal (i.e., their dot product is zero for any two distinct vectors). If they are not orthogonal, a more complex method involving matrix operations (like the formula P = A(A^TA)^-1A^T, where A's columns are the basis vectors) is required, often preceded by a Gram-Schmidt orthonormalization process.

Geometric Intuition

Think of a point on a wall. If you shine a flashlight directly at the wall, the shadow of the point is its orthogonal projection onto the wall. If the vector v represents the position of an object, and the subspace W represents a plane, the orthogonal projection of v onto W is the point on the plane closest to the object.

Why is Orthogonal Projection Important? (Applications)

Orthogonal projection is a cornerstone of many mathematical and computational techniques:

Least Squares Approximation: Finding the "best fit" line or curve for a set of data points is a classic application. The solution involves projecting the data vector onto the column space of the design matrix.
Data Analysis and Dimensionality Reduction: Techniques like Principal Component Analysis (PCA) use orthogonal projections to reduce the number of variables in a dataset while retaining as much variance as possible.
Computer Graphics: Used in rendering 3D objects onto a 2D screen, creating shadows, and other visual effects.
Signal Processing: Decomposing signals into orthogonal components, filtering noise, and data compression.
Machine Learning: Feature extraction and understanding relationships between data points.

How to Use This Calculator

Using this calculator is straightforward:

Enter Vector v: Input the components of your vector v as comma-separated numbers (e.g., 1,2,3).
Enter Subspace Basis Vectors u: For the subspace, provide one or more basis vectors. Each vector should be on a new line, with its components comma-separated (e.g.,
1,0,0
0,1,0).
Important Note: This calculator assumes the basis vectors you provide for the subspace are mutually orthogonal. If they are not, the result will be the sum of individual projections, which is generally not the true orthogonal projection onto the subspace.
Click "Calculate Projection": The result will display the components of the projected vector.

Example Calculation

Let's project vector v = [5, 6, 7] onto the subspace spanned by u1 = [1, 0, 0] and u2 = [0, 1, 0].

Projection onto u1:
v · u1 = (5*1) + (6*0) + (7*0) = 5
||u1||² = (1*1) + (0*0) + (0*0) = 1
proj_u1(v) = (5 / 1) * [1, 0, 0] = [5, 0, 0]
Projection onto u2:
v · u2 = (5*0) + (6*1) + (7*0) = 6
||u2||² = (0*0) + (1*1) + (0*0) = 1
proj_u2(v) = (6 / 1) * [0, 1, 0] = [0, 6, 0]
Total Projection (since u1 and u2 are orthogonal):
proj_W(v) = [5, 0, 0] + [0, 6, 0] = [5, 6, 0]

This result makes intuitive sense: projecting [5, 6, 7] onto the XY-plane (spanned by [1,0,0] and [0,1,0]) effectively removes the Z-component, resulting in [5, 6, 0].

orthogonal projection calculator