1. GeometryThe vector projection, also known as the vector component or vector resolution, of a vector a on (or onto) a non-zero vector b is the orthogonal projection of a onto a straight line parallel to b. The projection of a onto b is often written as proj𝐛𝐚 or a∥b. The vector component or vector resolute of a perpendicular to b, sometimes also called the vector rejection of a from b (denoted oproj𝐛𝐚 or a⊥b), is the orthogonal projection of a onto the plane (or, in general, hyperplane) that is orthogonal to b. Since both proj𝐛𝐚 and oproj𝐛𝐚 are vectors, and their sum is equal to a, the rejection of a from b is given by: oproj𝐛𝐚=𝐚−proj𝐛𝐚. To simplify notation, this article defines 𝐚1:=proj𝐛𝐚 and 𝐚2:=oproj𝐛𝐚. Thus, the vector 𝐚1 is parallel to 𝐛, the vector 𝐚2 is orthogonal to 𝐛, and 𝐚=𝐚1+𝐚2. The projection of a onto b can be decomposed into a direction and a scalar magnitude by writing it as 𝐚1=a1𝐛̂ where a1 is a scalar, called the scalar projection of a onto b, and b̂ is the unit vector in the... from wikipedia.org