THE DOT AND CROSS PRODUCTS OF TWO VECTORS
-----------------------------------------

Some people in class STILL seem to be having trouble with these so let
me do it in extreme detail.

An n-VECTOR is a n-tuple of numbers.
Example 3-vector:   (6,3,1).
Example 2-vector:   (x,y).

A SCALAR is an ordinary number.  Example: -9.

In formulas denote vector quantities with an ARROW above them.
(So you know "a" in the formula is a vector, not a scalar,
immediately, because you see that arrow.)


LEGAL THINGS TO DO WITH VECTORS AND SCALARS:
(Let capital letters be vectors and lowercase ones be scalars, since
my typewriter has no arrows.)

s V   <-- multiply scalar times vector. Example 2(1,2,3)=(2,4,6).
V s     Same thing.

A + B    add two vectors.  Example    (1,2,3)+(4,4,4)=(5,6,7).
A - B    subtract two vectors.  Example    (1,2,3)-(4,4,4)=(-3,-2,-1).

A . B   dot product of 2 vectors. Result is a SCALAR.
                 Example (1,2,3).(5,6,7) = 5+12+21 = 38.

A X B   cross product of two 3-vectors. Result is a VECTOR.

|A|   = length of a vector A = square root of (A.A)

|A-B| = distance between two vectors A and B = length of A-B



ILLEGAL BAD BAD BAD THINGS TO NOT DO:

A B   vector-vector multiplication without a dot or a cross. Meaningless.

cos(A)  try to take cosine of a vector!!???

A + s    try to add a vector and a scalar??!!

A + B    where A and B are different-dimensioned vectors like (1,2)+(5,4,2,8) ??!




PROPERTIES OF DOT AND CROSS PRODUCTS:

distributive associative and commutative laws hold, allowing
vector formulas to be simplified just like usual...

A X (B+C) = AXB + AXC

A . (B+C) = A.B + A.C

(A X B) X C = A X (B X C)

A.B = B.A

*EXCEPT* that

A X B = - B X A     ANTIcommutativity of cross product!! Watch out for that!



HOW TO COMPUTE DOT AND CROSS PRODUCTS:

(x,y,z) . (X,Y,Z) = xX + yY + zZ   dot product

(a,b,c) X (A,B,C) = (bC-cB, Ac-aC, aB-Ab)  cross product.
Note the middle term is reverse sign of the pattern in the outer two
terms...   Easiest way to remember this (?)

     i j k
     a b c
     A B C

where you get 6 terms:
       the \ diagonal terms with + signs:  ibC + jcA + kAB
   and the / diagonal terms with - signs: -icB - jaC - kbA



PROPERTIES:

A . (B X C) = B . (C X A) = C . (A X B)

A . (A X B) = 0            B . (A X B) = 0

  A.B 
------- = cos(angle between vectors A, B viewed from 0)
|A| |B|

|A X B| 
------- = sin(angle between vectors A, B viewed from 0)
|A| |B|

 A.B = 0           if A,B perpendicular
 A X B = (0,0,0)   if A,B parallel