Final answer:
Orthogonal projection is a concept in linear algebra used to project a point or vector onto a subspace, reducing dimensions while maintaining distances.
It involves calculating the closest vector within the subspace to the original vector.
Step-by-step explanation:
The term orthogonal projection can be visualized as the shadow a figure casts on a flat surface when illuminated by a light source positioned perpendicular to that surface. In more formal terms, the orthogonal projection of a vector a onto a line or plane is the vector b that lies in that line or plane, and is the closest to a.
To compute an orthogonal projection of a vector a onto a vector b, you use the formula projb(a) = ( (a · b) / (b · b) ) * b, where · represents the dot product.
The numerator finds the scalar projection of a onto b, and the denominator scales it to adjust for the length of b, to ensure that the resulting vector is in the direction of b but its length is such that it lies precisely on the line spanned by b.
This concept is widely used in various areas such as computer graphics, machine learning, and physics to simplify complex problems by reducing the dimensions while retaining the essential structure of the data or phenomena.