Scalar projection

In mathematics, the scalar projection of a vector \( \mathbf {a} \) on (or onto) a vector \( \mathbf {b} \) , also known as the scalar resolute of \( \mathbf {a} \) in the direction of \( \mathbf {b} \) , is given by:

\( {\displaystyle s=\left\|\mathbf {a} \right\|\cos \theta =\mathbf {a} \cdot \mathbf {\hat {b}} ,} \)

where the operator ⋅ {\displaystyle \cdot } \cdot denotes a dot product, \( {\hat {{\mathbf {b}}}} \) is the unit vector in the direction of \( \mathbf {b} \) , \( {\displaystyle \left\|\mathbf {a} \right\|} \) is the length of \( \mathbf {a}\) , and \( \theta \) is the angle between \( \mathbf {a} \) and \( \mathbf {b} \) .

If 0° ≤ θ ≤ 90°, as in this case, the scalar projection of a on b coincides with the length of the vector projection.

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 a {\displaystyle \mathbf {a} } \mathbf {a} on b {\displaystyle \mathbf {b} } \mathbf {b} , with a negative sign if the projection has an opposite direction with respect to b {\displaystyle \mathbf {b} } \mathbf {b} .

Vector projection of a on b (a1), and vector rejection of a from b (a2).

Multiplying the scalar projection of \( \mathbf {a} \) on \( \mathbf {b} \) by \( {\mathbf {{\hat b}}} \) converts it into the above-mentioned orthogonal projection, also called vector projection of \( \mathbf {a} \) on \( \mathbf {b} \).

Definition based on angle θ

If the angle \( \theta \) between \( \mathbf {a} \) and \( \mathbf {b} \) is known, the scalar projection of \( \mathbf {a} \) on \( \mathbf {b} \) can be computed using

\( {\displaystyle s=\left\|\mathbf {a} _{1}\right\|} {\displaystyle s=\left\|\mathbf {a} _{1}\right\|} \) in the figure)

Definition in terms of a and b

When \( \theta \) is not known, the cosine of \( \theta \) can be computed in terms of \( \mathbf {a} \) and \( \mathbf {b} \), by the following property of the dot product \( {\mathbf {a}}\cdot {\mathbf {b}} \):

\( {\displaystyle {\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {a} \right\|\,\left\|\mathbf {b} \right\|}}=\cos \theta \,} \)

By this property, the definition of the scalar projection \( s\, \) becomes:

\( {\displaystyle s=\left\|\mathbf {a} _{1}\right\|=\left\|\mathbf {a} \right\|\cos \theta =\left\|\mathbf {a} \right\|{\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {a} \right\|\,\left\|\mathbf {b} \right\|}}={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|}}\,} \)

Properties

The scalar projection has a negative sign if 90 < θ ≤ 180 {\displaystyle 90<\theta \leq 180} 90<\theta \leq 180 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 a 1 {\displaystyle \mathbf {a} _{1}} \mathbf {a} _{1} and its length ‖ a 1 ‖ {\displaystyle \left\|\mathbf {a} _{1}\right\|} {\displaystyle \left\|\mathbf {a} _{1}\right\|}:

\( {\displaystyle s=\left\|\mathbf {a} _{1}\right\|} \) if \( 0<\theta \leq 90 \) degrees,
\( {\displaystyle s=-\left\|\mathbf {a} _{1}\right\|} \) if \( 90<\theta \leq 180 \) degrees.