Scalar projection

In mathematics, the scalar projection of a vector formula_1 on (or onto) a vector formula_2, also known as the scalar resolute of formula_1 in the direction of formula_2, is given by:
where the operator formula_6 denotes a dot product, formula_7 is the unit vector in the direction of formula_2, formula_9 is the length of formula_1, and formula_11 is the angle between formula_1 and formula_2.
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 formula_1 on formula_2, with a negative sign if the projection has an opposite direction with respect to formula_2.
Multiplying the scalar projection of formula_1 on formula_2 by formula_19 converts it into the above-mentioned orthogonal projection, also called vector projection of formula_1 on formula_2.
Definition based on angle "θ".
If the angle formula_11 between formula_1 and formula_2 is known, the scalar projection of formula_1 on formula_2 can be computed using
Definition in terms of a and b.
When formula_11 is not known, the cosine of formula_11 can be computed in terms of formula_1 and formula_2, by the following property of the dot product formula_33:
By this property, the definition of the scalar projection formula_35 becomes:
Properties.
The scalar projection has a negative sign if formula_37. 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 formula_38 and its length formula_39: