1-page summary sheet:

Review of Vectors

$\bullet$ 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 $\Vec{A}$, sometimes written $\vec{\bf A}$ or $\vec{A}$ 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.

 

$\bullet$ Unit Vectors: In Cartesian coordinates (x,y,z) a vector $\Vec{A}$ can be expressed in terms of its three scalar components Ax,Ay,Az and the corresponding unit vectors $\iH, \jH, \kH$ (sometimes written as $\xH, \yH, \zH$ or occasionally as $\xH_1, \xH_2, \xH_3$) thus:

 \begin{displaymath}\Vec{A} = \iH A_x + \jH A_y + \kH A_z
\end{displaymath} (1)

where the little "hat" over a symbol means (in this context) that it has unit magnitude and thus imparts only direction to a scalar like Ax. 1

A unit vector $\Hat{a}$ can be formed from any vector $\Vec{a}$ by dividing it by its own magnitude a:

\begin{displaymath}\Hat{a} = {\Vec{a} \over a} \qquad \hbox{\rm where ~ }
a = \vert\Vec{a}\vert = \sqrt{a_x^2 + a_y^2 + a_z^2} \; .
\end{displaymath} (2)

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:

 

$\bullet$ Multiplying or Dividing a Vector by a Scalar: Multiplying a vector $\Vec{A}$ 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: $\Vec{A} b = b \Vec{A}$.

 

$\bullet$ 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,
\epsfig{file=PS/2vectAdd.ps,height=1.5in}
so we fall back on the algebraic method of adding components. Given $\Vec{A}$ from Eq. (1) and

 \begin{displaymath}\Vec{B} = \iH B_x + \jH B_y + \kH B_z
\end{displaymath} (3)

we write

 \begin{displaymath}\Vec{A} + \Vec{B} = \iH (A_x + B_x) + \jH (A_y + B_y) + \kH (A_z + B_z) \; .
\end{displaymath} (4)

Subtracting $\Vec{B}$ from $\Vec{A}$ is the same thing as adding $-\Vec{B}$.

 

$\bullet$ Multiplying Two Vectors . . .

. . .
to get a Scalar: we just add together the products of the components,

 \begin{displaymath}\Vec{A} \cdot \Vec{B} = A_x B_x + A_y B_y + A_z B_z
\end{displaymath} (5)

also known as the "dot product", which commutes: $\Vec{A} \cdot \Vec{B} = \Vec{B} \cdot \Vec{A}$.

. . .
to get a Pseudovector:
 
$\displaystyle \Vec{A} \times \Vec{B}$ = i (AyBz - AzBy) (6)
+ j (AzBx - AxBz)
+ k (AxBy - AyBx)

This "cross product" is actually a pseudovector (or, more generally, a tensor), because (unlike the nice dot product) it has the unsettling property of not commuting ( $\Vec{A} \times \Vec{B} = - \Vec{B} \times \Vec{A}$) but we often treat it like just another vector.
\epsfig{file=PS/vectProd.ps,height=1.5in}

We are going to be using these things continually throughout the rest of the course, so make sure you are adept with them.



Jess H. Brewer - Last modified: Fri Dec 9 12:41:19 PST 2016