Implement Orthogonal Projection of a Vector onto a Line

Problem

Compute the orthogonal projection of a vector onto a line defined by a direction vector. The projection gives the component of the vector that lies along the given direction.

Solution

import numpy as np

def orthogonal_projection(v, L):
    v = np.array(v, dtype=float)
    L = np.array(L, dtype=float)

    scalar = np.dot(v, L) / np.dot(L, L)
    projection = scalar * L

    return projection.tolist()

Explanation

1. The projection of vector v onto direction L is given by: proj_L(v) = (v . L / L . L) * L.
2. Compute the dot product of v and L for the numerator.
3. Compute the dot product of L with itself for the denominator.
4. Multiply the scalar ratio by L to get the projection vector.
5. The result lies along the direction of L and is the closest point on the line to v.

Complexity

• Time: O(n) where n is the dimensionality of the vectors
• Space: O(n) for the output projection vector