In mathematics, the scalar projection of a vector a on (or onto) a vector b , also known as the scalar resolute of a in the direction of b , is given by:
s = | a | cos θ = a ⋅ b ^ , where the operator ⋅ denotes a dot product, b ^ is the unit vector in the direction of b , | a | is the length of a , and θ is the angle between a and 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 a on b , with a negative sign if the projection has an opposite direction with respect to b .
Multiplying the scalar projection of a on b by b ^ converts it into the above-mentioned orthogonal projection, also called vector projection of a on b .
If the angle θ between a and b is known, the scalar projection of a on b can be computed using
s = | a | cos θ . (
s = | a 1 | in the figure)
Definition in terms of a and b
When θ is not known, the cosine of θ can be computed in terms of a and b , by the following property of the dot product a ⋅ b :
a ⋅ b | a | | b | = cos θ By this property, the definition of the scalar projection s becomes:
s = | a 1 | = | a | cos θ = | a | a ⋅ b | a | | b | = a ⋅ b | b | The scalar projection has a negative sign if 90 < θ ≤ 180 degrees. 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 a 1 and its length | a 1 | :
s = | a 1 | if
0 < θ ≤ 90 degrees,
s = − | a 1 | if
90 < θ ≤ 180 degrees.
