Euclidean norm in the context of Vector component

The gravity of Earth, denoted by g, is the net acceleration that is imparted to objects due to the combined effect of gravitation (from mass distribution within Earth) and the centrifugal force (from the Earth's rotation).It is a vector quantity, whose direction coincides with a plumb bob and strength or magnitude is given by the norm ${\displaystyle g=\|{\mathit {\mathbf {g} }}\|}$.

In SI units, this acceleration is expressed in metres per second squared (in symbols, m/s or m·s) or equivalently in newtons per kilogram (N/kg or N·kg). Near Earth's surface, the acceleration due to gravity, accurate to 2 significant figures, is 9.8 m/s (32 ft/s). This means that, ignoring the effects of air resistance, the vertical component of velocity of an object falling freely will increase in the downwards direction by about 9.8 metres per second (32 ft/s) every second.

In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Euclidean vectors can be added and scaled to form a vector space. A vector quantity is a vector-valued physical quantity, including units of measurement and possibly a support, formulated as a directed line segment. A vector is frequently depicted graphically as an arrow connecting an initial point A with a terminal point B, and denoted by ${\textstyle {\stackrel {\longrightarrow }{AB}}.}$

A vector is what is needed to "carry" the point A to the point B; the Latin word vector means 'carrier'. It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from A to B. Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operations which obey the familiar algebraic laws of commutativity, associativity, and distributivity. These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space.

In mathematics, the magnitude or size of a mathematical object is a property which determines whether the object is larger or smaller than other objects of the same kind. More formally, an object's magnitude is the displayed result of an ordering (or ranking) of the class of objects to which it belongs. Magnitude as a concept dates to Ancient Greece and has been applied as a measure of distance from one object to another. For numbers, the absolute value of a number is commonly applied as the measure of units between a number and zero.

In vector spaces, the Euclidean norm is a measure of magnitude used to define a distance between two points in space. In physics, magnitude can be defined as quantity or distance. An order of magnitude is typically defined as a unit of distance between one number and another's numerical places on the decimal scale.

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 ${\displaystyle \mathbf {b} }$.