Math 1C: Calculus III
Part 1: Multidimensional Differential Calculus
Lesson 4: The Cross Product
©Jeffrey A. Anderson

Key Terms List: Briggs Section 11.4
□ Component Form of Cross product of two vector (Lesson 4: Jeff’s Handwritten Notes)
□ Sine formula for cross product (p. 792)
□ Geometry of the Cross Product (p. 792)
	□ Using the cross product to determine if two vectors are parallel (p. 792)
□ Algebraic Properties of Cross Product (p. 793)
	□ Anti-commutativity of cross product: (OverVector[u])×(OverVector[v]) = - (OverVector[v])×(OverVector[u]) 
	□ Homogeneity of cross products: ab ((OverVector[u])×(OverVector[v])) = (ab (OverVector[u]))×(OverVector[v]) = (OverVector[u])×(ab (OverVector[v])) = (a (OverVector[u]))×(b (OverVector[v])) for a,b ∈ ℝ
	□ Left distributivity of cross product: (OverVector[u])×((OverVector[v]) +(OverVector[w])) = (OverVector[u])×(OverVector[v]) +(OverVector[u])×(OverVector[w])
	□ Right distributivity of cross product: ((OverVector[u])+(OverVector[v]))×(OverVector[w]) = (OverVector[u])×(OverVector[w]) +(OverVector[v])×(OverVector[w])
□ Thm 11.5: Cross product of standard basis vectors (OverVector[i]), (OverVector[j]), (OverVector[k]) (p. 794)
□ Torque and the cross product (p. 795)
□ Determinant of a 2×2 matrix (p. 794)
□ Determinant of a 3×3 matrix (p. 794)

Section 11.4: The Cross Product

Example 11.4.1 p. 793: Cross Product of Vectors in  (ℝ^3)

Let’s use Mathematica’s built-in Cross command to find the cross product given in Example 9.4.3 on p. 658. We begin by assuming that we have two vectors:

o = {0, 0, 0};
u = {1, 0.4, 1};
v = {0.5,1, 1.2};
p = {1.5,1.4,2.2};

xyu = {1, 0.4, 0};
xyv = {0.5,1, 0};
xyp = {1.5,1.25, 0} ;

xzu = {1, 0, 1};;
xzv = {0.5,0, 1.2};
xzp =  {1.5,0,2.2} ;

yzu = {0, 0.4, 1};
yzv = {0,1, 1.2};
yzp =  {0,1.4,2.2} ;

Show[
	Graphics3D[{PointSize[.01],Point[{u,v, p}]},Axes->True],(*end: Graphics3D command*)
	Graphics3D[{Blue,Thickness[0.006],Arrow[{{o,u}, {v,p}}]},Axes->True],
	Graphics3D[{Red, Thickness[0.006],Arrow[{{o,v}, {u,p}}]},Axes->True],	

	Graphics3D[{PointSize[.004],Point[{xyu,xyv,xyp}]},Axes->True],(*end: Graphics3D command*)
	Graphics3D[{Blue, Thickness[0.003],Line[{{o,xyu} , {xyp, xyv}}]},Axes->True],
	Graphics3D[{Red, Thickness[0.003],Line[{{xyu,xyp} , {xyv, o}}]},Axes->True],
	
	Graphics3D[{PointSize[.005],Point[{xzu,xzv,xzp}]},Axes->True],(*end: Graphics3D command*)
	Graphics3D[{Blue, Thickness[0.003],Line[{{o,xzu} , {xzp, xzv}}]},Axes->True],
	Graphics3D[{Red, Thickness[0.003],Line[{{xzu,xzp} , {xzv, o}}]},Axes->True],	


	Graphics3D[{PointSize[.005],Point[{yzu,yzv,yzp}]},Axes->True],(*end: Graphics3D command*)
	Graphics3D[{Blue, Thickness[0.003],Line[{{o,yzu} , {yzp,yzv}}]},Axes->True],
	Graphics3D[{Red, Thickness[0.003],Line[{{yzu,yzp} , {yzv, o}}]},Axes->True],

	
	ParametricPlot3D[{{t,0,0}, {0,t,0}, {0,0,t}},{t,-5,5}],
	TicksStyle->Directive[FontSize-> 12],
	AxesLabel->{x,y,z},
	PlotRange->{{-1,2.5},{-1,2.5},{-1,2.5}},
	ImageSize-> Full
](*end: Show command*)

.

Then, to calculate the cross product (OverVector[a])×(OverVector[b]) we simply enter the following code:

Cross[{a,b,c},{x,y,z}]

Notice, the command above produces a symbolic output. However, let’s change our input vectors to find a numeric answer for the vectors given in our target example:

a = {1,0,0};
b = {0,0, 1};

Cross[b,a]

Notice this is the same output as claimed in our textbook.