211 Scalar Projection
A scalar measuring how much of one vector lies along another that is used to extract the parallel part of a vector.
definition (Scalar Projection) The signed magnitude of the vector projection of \(\mathbf{b}\) onto \(\mathbf{a}\), also referred to as the component of \(\mathbf{b}\) along \(\mathbf{a}\), denoted by \(\text{comp}_{\mathbf{a}} \mathbf{b}\). It represents the length of the “shadow” and is positive if the angle between the vectors is acute and negative if it is obtuse. It is defined by the following formulas:
- \(\text{comp}_{\mathbf{a}} \mathbf{b} = \dfrac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}|}\)
- \(\text{comp}_{\mathbf{a}} \mathbf{b} = |\mathbf{b}| \cos \theta\)
where
- \(\mathbf{a}, \mathbf{b}\) are vectors in a coordinate system.
- \(\theta\) is the angle between vectors \(\mathbf{a}\) and \(\mathbf{b}\).
- \(|\mathbf{b}|\) is the magnitude of vector \(\mathbf{b}\).