Vector projection on a plane in the context of Vector (geometry)

The vector projection (also known as the vector component or vector resolution) of a vector a on (or onto) a nonzero 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 ${\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} }$ 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 ${\displaystyle \operatorname {oproj} _{\mathbf {b} }\mathbf {a} }$ or a_⊥b), is the orthogonal projection of a onto the plane (or, in general, hyperplane) that is orthogonal to b. Since both ${\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} }$ and ${\displaystyle \operatorname {oproj} _{\mathbf {b} }\mathbf {a} }$ are vectors, and their sum is equal to a, the rejection of a from b is given by: $${\displaystyle \operatorname {oproj} _{\mathbf {b} }\mathbf {a} =\mathbf {a} -\operatorname {proj} _{\mathbf {b} }\mathbf {a} .}$$