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Scalars and Vectors

An ordinary number that does not change when the coordinate frame changes is called a scalar. Multiplication of a vector by a scalar rescales the vector by multiplying each of its components as a special case of this rule:

$\displaystyle a \vA = a(A_x \hx + A_y\hy + A_z\hz) = (aA_x) \hx + (aA_y) \hy +
(aA_z) \hz $

Note well that the vector components $ A_x, A_y, A_z$ are themselves scalars. Indeed, we build a vector in the first place by taking a unit vector (of length one, ``pure direction'') and scaling it by its component length, e.g. $ A_x \hx$ , and then summing the vectors that make up its components!

The multiplication of a vector by a scalar is commutative:

$\displaystyle a \vA = \vA a $

and distributive.

$\displaystyle a(\vA + \vB) = a\vA + a\vB $

Robert G. Brown 2017-07-11