In mathematics, the scalar projection of a vector on (or onto) a vector also known as the scalar resolute of in the direction of is given by:
where the operator denotes a dot product, is the unit vector in the direction of is the length of and is the angle between and .[1]
The term scalar component refers sometimes to scalar projection, as, in Cartesian coordinates, the components of a vector are the scalar projections in the directions of the coordinate axes.
The scalar projection is a scalar, equal to the length of the orthogonal projection of on , with a negative sign if the projection has an opposite direction with respect to .
Multiplying the scalar projection of on by converts it into the above-mentioned orthogonal projection, also called vector projection of on .
If the angle between and is known, the scalar projection of on can be computed using
The formula above can be inverted to obtain the angle, θ.
When is not known, the cosine of can be computed in terms of and by the following property of the dot product :
By this property, the definition of the scalar projection becomes:
The scalar projection has a negative sign if . It coincides with the length of the corresponding vector projection if the angle is smaller than 90°. More exactly, if the vector projection is denoted and its length :