Vector Notation: a vector quantity is one that has both magnitude and direction. Another (equivalent) way of putting it is that a vector quantity has several components in orthogonal (perpendicular) directions. The idea of a vector is very abstract and general; one can define useful vector spaces of many sorts, some with an infinite number of orthogonal basis vectors, but the most familiar types are simple 3-dimensional quantities like position, speed, momentum and so on. The conventional notation for a vector is , sometimes written or or A but most clearly recognizable when in boldface with a little arrow over the top. On the blackboard a vector may be written with a tilde underneath, which is hard to generate in LATEX.
In Cartesian coordinates (x,y,z) a vector
expressed in terms of its three scalar components
and the corresponding unit vectors
(sometimes written as
A unit vector
can be formed from
by dividing it by its own magnitude a:
Already we have used a bunch of concepts before defining them properly, the usual chicken-egg problem with mathematics. Let's try to catch up:
Multiplying or Dividing a Vector by a Scalar: Multiplying a vector by a scalar b has no effect on the direction of the result (unless b=0) but only on its magnitude and/or the units in which it is measured - if b is a pure number, the units stay the same; but multiplying a velocity by a mass (for instance) produces an entirely new quantity, in that case the momentum.
Dividing a vector by a scalar c is the same as multiplying it by 1/c.
This type of product always commutes: .
Adding or Subtracting Vectors:
In two dimensions one can draw simple diagrams depicting
"tip-to-tail" or "parallelogram law" vector addition
(or subtraction); this is not so easy in 3 dimensions,
so we fall back on the algebraic method of adding components. Given from Eq. (1) and
Multiplying Two Vectors . . .
We are going to be using these things continually throughout the rest of the course, so make sure you are adept with them.