Scalar component

The vector projection of a vector a on (or onto) a nonzero vector b (also known as the vector component or vector resolution 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 ${\displaystyle a_{1}}$ 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

Typically, a vector projection is denoted in a bold font (e.g. a₁), and the corresponding scalar projection with normal font (e.g. a₁). In some cases, especially in handwriting, the vector projection is also denoted using a diacritic above or below the letter (e.g., ${\displaystyle {\vec {a}}_{1}}$ or a₁; see Euclidean vector representations for more details).

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