216 Vector Projection
A vector along a given direction that is the parallel part of another vector that is used to split a vector into parallel and remaining parts.
definition (Vector Projection) The vector representing the “shadow” of a vector \(\mathbf{b}\) cast onto the line containing a vector \(\mathbf{a}\), denoted by \(\text{proj}_{\mathbf{a}} \mathbf{b}\). Geometrically, it is the vector from the common initial point to the foot of the perpendicular dropped from the tip of \(\mathbf{b}\) onto the line containing \(\mathbf{a}\). It is calculated using the following formula:
- \(\text{proj}_{\mathbf{a}} \mathbf{b} = \left( \dfrac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}|^2} \right) \mathbf{a}\)
where
- \(\mathbf{a}, \mathbf{b}\) are vectors in a coordinate system.
- \(\mathbf{a} \cdot \mathbf{b}\) is the dot product of the two vectors.
- \(|\mathbf{a}|\) is the magnitude of vector \(\mathbf{a}\).