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 or scalar component 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 scalar projection is a scalar, equal to the length of the orthogonal projection of on , with a minus 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

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 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 and its length :

if degrees,
if degrees.

See also