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 :
Strang, Gilbert (2016). Introduction to linear algebra (5thed.). Wellesley: Cambridge press. ISBN978-0-9802327-7-6.
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