%% Vectors


%% Entering vectors in MATLAB
% 
% A vector in three space is represented in MATLAB as a 1 x 3 array
% (an array with with one row, three columns).  So, to enter the vectors  
% *a* = -3 *i* - 4 *j*- *k*, *b* = 6 *i* + 2 *j* + 3 *k* and 
% *u* = x *i* + y *j* + z *k*
% you type

a = [-3, -4, -1]

%%
b = [6, 2, 3]

%%
% and
syms x y z
u = [x, y, z]

%% 
% The command *syms* is necessary to tell MATLAB that x,y,z are symbolic.
% If you forget to do that, you will get an error message telling you that
% you have an undefined function or variable.

%% Comments on style
% When you publish your Mfile, MATLAB will list all the commands in a cell
% together, followed by all the output.  You create a new cell by starting a
% line with %%.  Make frequent use of these.  
%
% A line beginning with % indicates a comment.  To have the comments look
% nice in the published file, precede them with %%.
% Otherwise, they come out as lines starting with %, as this shows:

x^2

% Here's a comment.

%% Vector arithmetic
%
% You can add and subtract these and multiply them by scalars.

a + b

%%

5*a

%% Dot product
% You can calculate the dot product of two vectors with the command *dot*.

dot(a,b)

%%
dot(a,u)

%% 
dot(u,a)

%%  
% This isn't the same as dot(a,u).  MATLAB doesn't know that x, y and z are
% real.  It assumes they are complex.  It takes the complex conjugate of
% the components of the first vector.  The complex conjugate of the complex
% number c+di is c-di where i is the square root of -1.  So the
% complex conjugate of 3+7i is 3-7i. Since we are only dealing with
% real vectors, you can get around this problem by defining a function
% *realdot*

realdot = @(u,v) u*transpose(v)

%%
realdot(u,a)

%%
% or if you downloaded the Mfiles, you can just give the command *realdot*.
%%
%  Or you could tell MATLAB the variables are real by giving the command
syms x y z real
dot(u,a)
%% Cross product
% You can calculate the cross product of two vectors with the command
% *cross*.

cross(a,b)

%%
cross(b,a)

%%
cross(a,u)