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
Famous quotes containing the word projection:
“In the case of our main stock of well-worn predicates, I submit that the judgment of projectibility has derived from the habitual projection, rather than the habitual projection from the judgment of projectibility. The reason why only the right predicates happen so luckily to have become well entrenched is just that the well entrenched predicates have thereby become the right ones.”
—Nelson Goodman (b. 1906)