Scalar projection

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If 0° ≤ θ ≤ 90°, as in this case, the scalar projection of a on b coincides with the length of the vector projection.
Vector projection of a on b (a1), and vector rejection of a from b (a2).

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[edit]

If the angle between and is known, the scalar projection of on can be computed using

( in the figure)

Definition in terms of a and b[edit]

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:


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[edit]