Section 12.4: Cross Product

Review

Questions

What are the components of the vector that points from point (1,0,-2) to point (2,4,3)?

What is a unit vector that points in that same direction?

Questions

What is the geometric significance of scalar multiplication?

What is the geometric significance of vector addition?

What is the geometric significance of vector subtraction?

What is the geometric significance of the dot product?

Cross product definition

The dot product is a multiplication-like operation between two vectors (two 2D vectors or two 3D vectors) that gives a scalar value.

The cross product is a multiplication-like operation between two 3D that gives a vector value.

Geometric definition

The cross product of 3D vectors (Overscript[u, ->]) and (Overscript[v, ->]) is denoted (Overscript[u, ->])×(Overscript[v, ->]).

The magnitude of (Overscript[u, ->])×(Overscript[v, ->])  is given by                                                                                                    |(Overscript[u, ->])×(Overscript[v, ->])|=|(Overscript[u, ->])| |(Overscript[v, ->])|sin(θ)                                                                where θ is the smaller of the two angles between vectors (Overscript[u, ->]) and (Overscript[v, ->]).

The direction of (Overscript[u, ->])×(Overscript[v, ->]) is perpendicular to both (Overscript[u, ->]) and (Overscript[v, ->]). There are two such directions. Choose the one that satisfies the right-hand rule.

u={2,-1,3};
v={1,2,1};
ucrossv=Cross[u,v];
vcrossu=Cross[v,u];
mag=Sqrt[ucrossv.ucrossv];
Graphics3D[{Arrow[{{0,0,0},u}],
Arrow[{{0,0,0},v}],
Green,Arrow[{{0,0,0},vcrossu/mag}],
Text[Style["vxu (hat)",FontSize->18],1.1vcrossu/mag,{1,1}],
Blue,Arrow[{{0,0,0},ucrossv/mag}],
Text[Style["uxv (hat)",FontSize->18],1.1ucrossv/mag,{1,1}],
Text[Style["u",FontSize->18],1.1u,{1,1}],
Text[Style["v",FontSize->18],1.1v,{1,0}]
}
]

Right-hand rule

To determine the direction of (Overscript[u, ->])×(Overscript[v, ->]) from the two possible directions, using the fingers on your right hand

Point your index finger in the direction of (Overscript[u, ->]).

Sweep that finger towards the direction of (Overscript[v, ->]). Your thumb will point in the correct direction of (Overscript[u, ->])×(Overscript[v, ->]).

Questions

Let (Overscript[i, ->])=⟨1,0,0⟩, (Overscript[j, ->])=⟨0,1,0⟩, and (Overscript[k, ->])=⟨0,0,1⟩.

What is (Overscript[i, ->])×(Overscript[j, ->])?

Cross[{1,0,0},{0,1,0}]

What is (Overscript[i, ->])×(Overscript[k, ->])?

Cross[{1,0,0},{0,0,1}]

What is (Overscript[j, ->])×(Overscript[k, ->])?

Cross[{0,1,0},{0,0,1}]

Question

Are (Overscript[u, ->])×(Overscript[v, ->]) and (Overscript[v, ->])×(Overscript[u, ->]) the same?

Cross[{0,1,0},{0,0,1}]
Cross[{0,0,1},{0,1,0}]

Component definition

If (Overscript[u, ->])=⟨(Subscript[u, 1]),(Subscript[u, 2]),(Subscript[u, 3])⟩ and (Overscript[v, ->])=⟨(Subscript[v, 1]),(Subscript[v, 2]),(Subscript[v, 3])⟩, then
	(Overscript[u, ->])×(Overscript[v, ->])=⟨(Subscript[u, 2]) (Subscript[v, 3])-(Subscript[u, 3]) (Subscript[v, 2]),(Subscript[u, 3]) (Subscript[v, 1])-(Subscript[u, 1]) (Subscript[v, 3]),(Subscript[u, 1]) (Subscript[v, 2])-(Subscript[u, 2]) (Subscript[v, 1])⟩

Cross[{(Subscript[u, 1]),(Subscript[u, 2]),(Subscript[u, 3])},{(Subscript[v, 1]),(Subscript[v, 2]),(Subscript[v, 3])}]

So it’s products of the components of the vectors from the other dimensions (e.g. y,z) that figure in the component for x, say (indices 2 and 3 are “yz” indices, and so on).

And if we check the permutations of the indices, then 123, 231, and 312 give rise to positive contributions, while 321, 213, and 132 give rise to negative contributions.

Questions

Let (Overscript[i, ->])=⟨1,0,0⟩,   (Overscript[j, ->])=⟨0,1,0⟩,   and (Overscript[k, ->])=⟨0,0,1⟩.

What is (Overscript[i, ->])×(Overscript[j, ->])?

What is (Overscript[i, ->])×(Overscript[k, ->])?

What is (Overscript[j, ->])×(Overscript[k, ->])?

Question

What is a vector that is perpendicular to both (Overscript[u, ->])=⟨2,-1,3⟩ and (Overscript[v, ->])=⟨1,2,1⟩?

Cross[{2,-1,3},{1,2,1}]

Cross product geometric properties

Questions

What must be true about vectors (Overscript[u, ->]) and (Overscript[v, ->]) if (Overscript[u, ->])×(Overscript[v, ->])=0?

Place vectors (Overscript[u, ->]) and (Overscript[v, ->]) so there initial points are the same. They form two of the sides of a parallelogram. What is the area of this parallelogram in terms of the lengths of these vectors and the angle between them?

Clear[u,v];null={0,0,0};uu={2,-1,3};vv={1,2,1};uuUnit=uu/Sqrt[uu.uu];vvUnit=vv/Sqrt[vv.vv];
arc=ParametricPlot3D[.5(Sin[t] vvUnit+Cos[t] uuUnit),{t,0,Pi/2}];
Show[Graphics3D[{Arrow[{null,uu}],Arrow[{null,vv}],Dashing[{0.02}],Line[{uu,uu+vv,vv}],Text[Style[u,FontSize->18],.5uu,{1,1}],Text[Style[v,FontSize->18],.5vv,{1,0}],Text[Style[θ,FontSize->18],.5(uuUnit+vvUnit)]}],arc]

What is the area of the parallelogram formed from vectors (Overscript[u, ->])=⟨2,-1,3⟩ and (Overscript[v, ->])=⟨1,2,1⟩?

Norm[Cross[{2,-1,3},{1,2,1}]]

What is the area of the triangle with vertices (0,1,0), (2,-1,-1), (-1,0,1)?

1/2*Norm[Cross[{2,-1,-1}-{0,1,0},{-1,0,1}-{0,1,0}]]

Cross product algebraic properties

Question

How are (Overscript[a, ->])×(Overscript[b, ->]) and (Overscript[b, ->])×(Overscript[a, ->]) related? (The cross-product is non-commutative.)

Other properties

(s (Overscript[a, ->]))×(Overscript[b, ->])=s((Overscript[a, ->])×(Overscript[b, ->]))=(Overscript[a, ->])×(s (Overscript[b, ->]))      (where s is a scalar)

(Overscript[a, -> ])×((Overscript[b, ->])+(Overscript[c, ->]))=(Overscript[a, ->])×(Overscript[b, ->])+(Overscript[a, ->])×(Overscript[c, ->])

(Overscript[a, ->])·((Overscript[b, ->])×(Overscript[c, ->]))=((Overscript[a, ->])×(Overscript[b, ->]))·(Overscript[c, ->])

Questions

Show property 2 is true by computing each side using
	(Overscript[a, ->])=⟨(Subscript[a, 1]),(Subscript[a, 2]),(Subscript[a, 3])⟩, (Overscript[ b, ->])=⟨(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3])⟩,  (Overscript[c, ->])=⟨(Subscript[c, 1]),(Subscript[c, 2]),(Subscript[c, 3])⟩

Show that the cross-product is non-associative: that is, that 
	(Overscript[a, -> ])×((Overscript[b, ->])×(Overscript[c, ->]))≠((Overscript[a, ->])×(Overscript[b, ->]))×(Overscript[c, ->])										(in general).

What is (Overscript[k, ->])×((Overscript[i, ->])+2(Overscript[j, ->]))?

Cross[{0,0,1},{1,2,0}]

What is  (Overscript[a, ->])×(Overscript[b, ->])= ((Subscript[a, 1])(Overscript[i, ->])+(Subscript[a, 2])(Overscript[j, ->])+(Subscript[a, 3])(Overscript[k, ->]))×((Subscript[b, 1])(Overscript[i, ->])+(Subscript[b, 2])(Overscript[j, ->])+(Subscript[b, 3])(Overscript[k, ->]))?

Clear[a,b,c,d,e,f]
Cross[{a,b,c},{d,e,f}]