Math 1C with Jeff Anderson
Unit 1: Vectors and the Geometry of Space
©Jeffrey A. Anderson
Class I: Vectors in 2D

Key Terms List: Briggs Section 11.1
□ Vector (p. 757)
□ Vectors in (ℝ^2) (p. 757)
□ Tail (initial point) of a vector (p.757)
□ Head (terminal point) of a vector (p. 757)
□ Equivalent vectors (p. 758)
□ Vector operations (p. 758 - 760)
	□ Scalar-vector multiplication (p. 758)
	□ Vector addition (p. 759)
	□ Vector difference (p. 759)
	□ Triangle law (p. 759)
	□ Parallelogram law (p. 759)
	□ Magnitude/Length: 2-norm of a vector  in (ℝ^2) (p. 761)
□ Properties of Vector Operations (p. 764)
	□ Additive Identity: (OverVector[a]) + (OverVector[0]) =  (OverVector[a]) = (OverVector[0]) + (OverVector[a])
	□ Commutativity of vector addition: (OverVector[a])+(OverVector[b]) = (OverVector[b])+(OverVector[a])
	□ Distributivity of scalar-vector multiplication over vector addition: α ((OverVector[a])+(OverVector[b])) = α·(OverVector[a])+α·(OverVector[b])
	□ Associativity of scalar-scalar-vector multiplication: (α β) · (OverVector[a]) = α · (β · (OverVector[a]))
	□ Associativity of vector addition: ((OverVector[a]) + (OverVector[b])) + (OverVector[c]) = (OverVector[a]) + ((OverVector[b]) + (OverVector[c]))
	□ Additive Inverses: (OverVector[a]) +(-(OverVector[a])) = (OverVector[0]) = (-(OverVector[a])) + (OverVector[a])
	□ Distributivity of scalar addition and scalar-vector multiplication:  (α + β) · (OverVector[a])  =  α· (OverVector[a]) + β · (OverVector[a]) 
	□ Scalar-vector Multiplicative Identity: 1·(OverVector[a]) = (OverVector[a])  
□ Representations of a vector (p. 762 - 763)
	□ Geometric representation (p. 762)
	□ Linear combination of basis vectors (p. 763)

Section 11.1: Vectors in 2D

Example 1: Points in (ℝ^2)

??Table

p = {5,4};
q = {1,1};
Show[
	Graphics[{PointSize[.02],Point[{q,p}]},Axes->True],(*end: Graphics command*)
	Axes->True,
	AspectRatio->Automatic,
	Ticks-> {Table[i,{i,-1,6}], Table[i,{i,-1,6}]},
	TicksStyle->Directive[FontSize-> 16],
	PlotRange-> {{-1,6},{-1,6}}
](*end: Show command*)

Example 2: Vectors in (ℝ^2)

p = {5,4};
q = {1,1};
Show[
	Graphics[{PointSize[.01],Point[{q,p}]},Axes->True],(*end: Graphics command*)
	Graphics[{Thickness[.004],Arrow[{q,p}]},Axes->True],(*end: Graphics command*)
	Axes->True,
	AspectRatio->Automatic,
	Ticks-> {Table[i,{i,-1,6}], Table[i,{i,-1,6}]},
	TicksStyle->Directive[FontSize-> 16],
	PlotRange-> {{-1,6},{-1,6}}
](*end: Show command*)

Example 3: Vectors in (ℝ^2)

o = {0,0};
p = {3,1};
a = {-1,1};
q = a+p;
Show[
		Graphics[
			{Thickness[.004],Arrow[{o,p}],Arrow[{o,a}]},Axes->True
		],(*end Graphics command*)
		Axes->True,
		AspectRatio->Automatic,
		PlotRange-> {{-2,4},{-1,3}}
](*end Show command*)

Show[
	Graphics[
		{Thickness[.004],Arrow[{o,p}],Arrow[{o,a}]},
		Axes->True
	], (*end Graphics command*)

	Graphics[
		{Red,Thickness[.004],Arrow[{o,q}]},
		Axes->True
	],(*end Graphics command*)

	Graphics[
		{Blue,Dashed,Thickness[.004],Arrow[{a,q}], Arrow[{p,q}]},
		Axes->True
	], (*end Graphics command*)

	Axes->True,
	AspectRatio->Automatic,
	PlotRange-> {{-2,4},{-1,3}}
](*end Show command*)