Cross Product

MAT 229, Spring 2021

Week 16

Supporting materials

If you wish to get a different perspective on the notes below, try either of the following textbook sections.

Stewart’s Calculus
	Section 12.4: Cross product

Boelkins/Austin/Schlicker’s Active Multivariable Calculus
	Section 9.4: The cross product

Review

Questions

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

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

(Video)

Questions

What is the geometric significance of scalar multiplication? (Video)

What is the geometric significance of vector addition? (Video)

What is the geometric significance of vector subtraction? (Video)

What is the geometric significance of the dot product? (Video)

Cross product definition

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

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

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.

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, ->]).

(source)
Magnetic force is a cross product. Torque is a cross product. There are lots of cross products in physics!

Clear[u,v]
uu={2,-1,3};
vv={1,2,1};
ucrossv=Cross[uu,vv];
vcrossu=Cross[vv,uu];
mag=Sqrt[ucrossv.ucrossv];
Graphics3D[{Arrow[{{0,0,0},uu}],
Arrow[{{0,0,0},vv}],
Green,Arrow[{{0,0,0},vcrossu/mag}],
Text[Cross[v,u],1.1vcrossu/mag,{1,1}],
Blue,Arrow[{{0,0,0},ucrossv/mag}],
Text[Cross[u,v],1.1ucrossv/mag,{1,1}],
Text[u,1.1uu,{1,1}],
Text[v,1.1vv,{1,0}]
},BaseStyle->FontSize->18
]

Questions

Let (Overscript[i, ->])=⟨1,0,0⟩, (Overscript[j, ->])=⟨0,1,0⟩, and (Overscript[k, ->])=⟨0,0,1⟩. Use the above geometric definition to determine the following cross products.

Graphics3D[{Thick,Red,Arrow[{{0,0,0},{1,0,0}}],Text["(Overscript[i, ->])",{1.0,0.1,0.1}],Blue,Arrow[{{0,0,0},{0,1,0}}],Text["(Overscript[j, ->])",{.1,1.,.1}],Green,Arrow[{{0,0,0},{0,0,1}}],Text["(Overscript[k, ->])",{.1,.1,1}]},BaseStyle->{FontSlant->Italic,FontSize->18},Axes->True,AxesLabel->{x,y,z},PlotRange->{{0,1.5},{0,1.5},{0,1.5}}]

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}]

(Video)

Question

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

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])}]

Notes:

The first component of the cross product consists of the second and third components of the given two vectors.

The second component of the cross product consists of the first and third components of the given two vectors.

The third component of the cross product consists of the first and second components of the given two vectors.

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.

Since I hate memorizing stuff like all that above, I just think in terms of components, and the three basis vectors of our Cartesian coordinate system:

(Overscript[u, ->])×(Overscript[v, ->])= ((Subscript[u, 1])(Overscript[i, ->])+(Subscript[u, 2])(Overscript[j, ->])+(Subscript[u, 3])(Overscript[k, ->]))×((Subscript[v, 1])(Overscript[i, ->])+(Subscript[v, 2])(Overscript[j, ->])+(Subscript[v, 3])(Overscript[k, ->]))?
	= ((Subscript[u, 2]) (Subscript[v, 3])-(Subscript[u, 3]) (Subscript[v, 2]))(Overscript[i, ->])+((Subscript[u, 3]) (Subscript[v, 1])-(Subscript[u, 1]) (Subscript[v, 3]))(Overscript[j, ->])+((Subscript[u, 1]) (Subscript[v, 2])-(Subscript[u, 2]) (Subscript[v, 1]))(Overscript[k, ->])

Questions

Let (Overscript[i, ->])=⟨1,0,0⟩,   (Overscript[j, ->])=⟨0,1,0⟩,   and (Overscript[k, ->])=⟨0,0,1⟩. Use the above component definition to determine the following cross products. (Compare with the results using the geometric definition.)

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⟩?	(Video)

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

Cross product geometric properties

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, ->]).

If (Overscript[u, ->])×(Overscript[v, ->])=(Overscript[0, ->]), then its magnitude is 0. One of three things must be true.

(Overscript[u, ->])=(Overscript[0, ->]), or

(Overscript[v, ->])=(Overscript[0, ->]), or

sin(θ)=0 which means θ=0. The vectors must be parallel, pointing in the same direction or in the exact opposite direction.

Place vectors (Overscript[u, ->]) and (Overscript[v, ->]) so their initial points are the same. They form two of the sides of a parallelogram. The area of a parallelogram is height×base.

Graphics[{Line[{{0,0},{2,0},{2.5,1},{0.5,1},{0,0}}],Dashed,Arrow[{{0.8,-0.1},{0,-0.1}}],Arrow[{{1.2,-0.1},{2,-0.1}}],Text["Base",{1,-0.1}],Arrow[{{0.5,0.4},{0.5,0}}],Arrow[{{0.5,0.6},{0.5,1}}],Text["Height",{0.5,0.5}]}]

The area of this parallelogram is

|(Overscript[u, ->])| |(Overscript[v, ->])|sin(θ)=|(Overscript[u, ->])×(Overscript[v, ->])|

(in terms of the lengths of these vectors and the angle θ 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]

Questions

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

w=Cross[{2,-1,3},{1,2,1}]
Norm[w]
Sqrt[w.w]

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

(* define two vectors, making two legs of the triangle: *)
uu={2,-1,-1}-{0,1,0}
vv={-1,0,1}-{0,1,0}
(* A triangle is half of a parallelogram! *)
1/2*Norm[Cross[uu,vv]]

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])⟩

(Video)

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

(Video)

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

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

(Video)

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}]

(Video)

Homework

IMath problems on cross products.