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The vector projection of a vector a on a nonzero vector b, sometimes denoted proj b a {\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} } , is the orthogonal projection of a onto a straight line parallel to b. It is a vector parallel to b, defined as:
In turn, the scalar projection is defined as:
Which finally gives:
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 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: a 2 = a − a 1 . {\displaystyle \mathbf {a} _{2}=\mathbf {a} -\mathbf {a} _{1}.}