Orthogonal Projection

Project the vector v onto the line spanned by u.

proj_u(v) = ((v . u) / (u . u)) * u

The result is the point on the line through u closest to v, and v - proj is perpendicular to u. This is the whole idea behind least squares: fitting a linear model is projecting the target onto the span of the features.

If u is the zero vector the line is a point, so return a zero vector.

Function signature

def orthogonal_projection(v, u):

Example

Input

orthogonal_projection(np.array([3.0, 4.0]), np.array([1.0, 0.0]))

Output

array([3., 0.])

Reasoning

v . u = 3 and u . u = 1, so the projection is 3 * [1, 0] = [3, 0]. What is left, [0, 4], is perpendicular to u.

Constraints and notes

• If u is the zero vector, return a zero vector.