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 .
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 .
Definition based on angle θ If the angle between and is known, the scalar projection of on can be computed using
( in the figure)
The formula above can be inverted to obtain the angle, θ.
Definition in terms of a and b 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:
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 :
if if
See also Scalar product Cross product Vector projection
Sources Dot products - www.mit.org Scalar projection - Flexbooks.ck12.org Scalar Projection & Vector Projection - medium.com Lesson Explainer: Scalar Projection | Nagwa
References