In mathematics, the scalar projection of a vector \mathbf{a} on (or onto) a vector \mathbf{b}, also known as the scalar resolute of \mathbf{a} in the direction of \mathbf{b}, is given by: where the operator \cdot denotes a dot product, is the unit vector in the direction of \mathbf{b}, is the length of \mathbf{a}, and \theta is the angle between \mathbf{a} and \mathbf{b}. 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 \mathbf{a} on \mathbf{b}, with a negative sign if the projection has an opposite direction with respect to \mathbf{b}. Multiplying the scalar projection of \mathbf{a} on \mathbf{b} by converts it into the above-mentioned orthogonal projection, also called vector projection of \mathbf{a} on \mathbf{b}.

Definition based on angle θ

If the angle \theta between \mathbf{a} and \mathbf{b} is known, the scalar projection of \mathbf{a} on \mathbf{b} can be computed using The formula above can be inverted to obtain the angle, θ.

Definition in terms of a and b

When \theta is not known, the cosine of \theta can be computed in terms of \mathbf{a} and \mathbf{b}, by the following property of the dot product : By this property, the definition of the scalar projection s becomes:

Properties

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 :

Sources