Vector projection in the context of Vector (geometry)

The radial velocity or line-of-sight velocity of a target with respect to an observer is the rate of change of the vector displacement between the two points. It is formulated as the vector projection of the target-observer relative velocity onto the relative direction or line-of-sight (LOS) connecting the two points.

The radial speed or range rate is the temporal rate of the distance or range between the two points. It is a signed scalar quantity, formulated as the scalar projection of the relative velocity vector onto the LOS direction. Equivalently, radial speed equals the norm of the radial velocity, modulo the sign.

A spatial gradient is a gradient whose components are spatial derivatives, i.e., rate of change of a given scalar physical quantity with respect to the position coordinates in physical space. Homogeneous regions have spatial gradient vector norm equal to zero.When evaluated over vertical position (altitude or depth), it is called vertical derivative or vertical gradient; the remainder is called horizontal gradient component, the vector projection of the full gradient onto the horizontal plane.

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

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