Linear Independence of Vectors?

(* Two vectors in real 3-space that are linearly independent (one is not a multiple of hte other). *)
	a={1,2,3}; 
	b={2,-2,3}; 

(* A visual representation of each.  *)
	aVector = Graphics3D[{Red,Thick,Arrowheads[Large],Arrow[{{0,0,0},a}]}];
	bVector = Graphics3D[{Red,Thick,Arrowheads[Large],Arrow[{{0,0,0},b}]}];

(* Show the vectors together in 3-space.  *)
	Show[aVector,bVector,PlotRange->{{-5,5},{-5,5},{-5,5}},AxesOrigin->{0,0,0},Axes->True,AxesLabel->{x,y,z}]

(* Define the plane determined by the two vectors via using their cross product as the common normal vectoe.  *)
	abNormal = Cross[a,b];

(* Show the two vectors inside the plane that they determine. *)
	abPlane = ParametricPlot3D[{-(abNormal[[2]]*u+abNormal[[3]]*v)/abNormal[[1]],u,v},{u,-10,10},{v,-10,10},AxesOrigin->{0,0,0},Mesh->None,AspectRatio->1];
	Show[aVector,bVector,abPlane,PlotRange->{{-5,5},{-5,5},{-5,5}},AxesOrigin->{0,0,0},Axes->True,AxesLabel->{x,y,z}]

(* Create two new vectors, one in side the place determined by a abd b and one outside of it. *)
	c= -1*a + b; 
	d={3,1,2}; 
	
(*  Show all of the new elements together. *)
	cVector=Graphics3D[{Blue,Thick,Arrowheads[Large],Arrow[{{0,0,0}, c}]}];
	dVector =Graphics3D[{Green,Thick,Arrowheads[Large],Arrow[{{0,0,0},d}]}];
	Show[aVector,bVector,cVector,dVector,abPlane,PlotRange->{{-5,5},{-5,5},{-5,5}},AxesOrigin->{0,0,0},Axes->True,AxesLabel->{x,y,z}]

Equation of a line in 3 - space?

(* Two linear equations in three space, each determining a plane. *) 	
	plane1 = Plot3D[(4-x-2y)/3,{x,-10,10},{y,-10,10},AxesOrigin->{0,0,0},Mesh->None,AspectRatio->1.5,PlotRange->{-20,20}];
	plane2 = Plot3D[(1-2x+2y)/3,{x,-10,10},{y,-10,10},AxesOrigin->{0,0,0},Mesh->None,AspectRatio->1];

(* Two planes that are neither parallel nor the same plane will intersect in a line in real 3-space.  *)
	Show[plane1,plane2]

(* This is the parameterized line shown in class:  x(t) = (5-6t)/3, y(t) = (7-3t)/6, and z(t) = t.  *)
	plotIntersection = ParametricPlot3D[{5/3-2t,7/6-t/2,t},{t,-5,7},PlotStyle->{Thick, Red}];

(* See how a line is determined in real 3-space, either by two equations in the three space variables, or three equations when parameterizing the line by t. *)
	Show[plane1,plane2,plotIntersection, Framed->False,ViewPoint->{4,2,1},Boxed->False]