Vector operations

Here, all vectors are drawn starting from the origin.

A vector, defined by its x - and y - coordinates.

orig={0,0};
b={1,2};
c={3,1};
l[b_,x_,y_]:=Line[{orig,b,b+{x,y},{x,y},orig}];

Manipulate[
Graphics[Arrow[{orig,{x,y}}],Axes->True,PlotRange->{{-2,2},{-2,2}},ImageSize->200],{x,2,-2,-.1},{y,2,-2,-.1}]

Addition of vectors, visualized by the parallogram rule. The vector b (red) is fixed.

Manipulate[
Graphics[{l[b,x,y],Red,Arrow[{orig,b}],Green,Arrow[{orig,{x,y}}],Black,Thick,Arrow[{orig,b+{x,y}}]},AspectRatio->1,Axes->True,PlotRange->{{-4,4},{-4,4}},ImageSize->200],{x,2,-2,-.1},{y,2,-2,-.1}]

Linear combinations of vectors.

Manipulate[
Graphics[{Gray,Arrow[{orig,beta*c}],Gray,Arrow[{orig,alpha*b}],Red,Arrow[{orig,c}],Green,Arrow[{orig,b}],Black,Thick,Arrow[{orig,alpha*b+beta*c}]},AspectRatio->1,Axes->True,PlotRange->{{-6,6},{-6,6}},ImageSize->200],{alpha,2,-2,-.1},{beta,2,-2,-.1}]

A 3 D vector :

orig3={0,0,0};
Manipulate[
Graphics3D[{Green,Thick,Arrow[{orig3,{x,y,z}}]},Axes->True,PlotRange->{{-2,2},{-2,2},{-2,2}},AspectRatio->1,AxesLabel->{1,2,3},ImageSize->200],{x,2,-2,-.1},{y,2,-2,-.1},{z,2,-2,-0.1}]

The parallelogram rule works in 3 D as well :

b3={0,0,1};
MatrixForm[b3];

l3[b3_,x_,y_,z_]:=Line[{orig3,b3,b3+{x,y,z},{x,y,z},orig3}];
Manipulate[
Graphics3D[{l3[b3,x,y,z],Red,Arrow[{orig3,b3}],Green,Arrow[{orig3,{x,y,z}}],Black,Thick,Arrow[{orig3,b3+{x,y,z}}]},Axes->True,PlotRange->{{-2,2},{-2,2},{-2,2}},AspectRatio->1,AxesLabel->{1,2,3},ImageSize->200],{x,1,-1,-.1},{y,1,-1,-.1},{z,1,-1,0.1}]