Vector Projection

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 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 ab and ab, respectively.

Read more about Vector Projection:  Definitions in Terms of a and b, Matrix Representation, Uses, Generalizations

Famous quotes containing the word projection:

    Those who speak of our culture as dead or dying have a quarrel with life, and I think they cannot understand its terms, but must endlessly repeat the projection of their own desires.
    Muriel Rukeyser (1913–1980)