Scalar projection
In mathematics, the scalar projection of a vector
also known as the scalar resolute of
denotes a dot product,
is the unit vector in the direction of
θ
is the angle between
[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
, with a negative sign if the projection has an opposite direction with respect to
Multiplying the scalar projection of
converts it into the above-mentioned orthogonal projection, also called vector projection of
If the angle
θ
is known, the scalar projection of
can be computed using The formula above can be inverted to obtain the angle, θ.
θ
is not known, the cosine of
θ
can be computed in terms of
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
< θ ≤
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
