The vector projection of a vector a on (or onto) a nonzero vector b (also known as the vector component or vector resolute of a in the direction of b) is the orthogonal projection of a onto a straight line parallel to b. It is a vector parallel to b, defined as
where a1 is a scalar, called the scalar projection of a onto b, and b̂ is the unit vector in the direction of b. In turn, the scalar projection is defined as
where the operator · denotes a dot product, |a| is the length of a, and θ is the angle between a and b. The scalar projection is equal to the length of the vector projection, with a minus sign if the direction of the projection is opposite to the direction of b.
The vector component or vector resolute of a perpendicular to b, sometimes also called the vector rejection of a from b, is the orthogonal projection of a onto the plane (or, in general, hyperplane) orthogonal to b. Both the projection a1 and rejection a2 of a vector a are vectors, and their sum is equal to a, which implies that the rejection is given by
The vector projection of a on b and the corresponding rejection are sometimes denoted by a∥b and a⊥b, respectively.
Read more about Vector Projection: Definitions in Terms of a and b, Matrix Representation, Uses, Generalizations
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