Some CrossProduct Identities

> restart:with(linalg):

Warning, new definition for norm

Warning, new definition for trace

> U:=vector(3,[u1,u2,u3]);V:=vector(3,[v1,v2,v3]);W:=vector(3,[w1,w2,w3]);

> dotprod(U,V);innerprod(U,V);

> A:=matrix(2,2,[a,b,c,d]);Det(A)=det(A);

> B:=matrix(3,3,[a,b,c,u1,u2,u3,v1,v2,v3]);Det(B)=det(B);

> Xvector:=crossprod(U,V);

> innerprod(U,Xvector);innerprod(V,Xvector);

> u:=vector(3,[1,-2,3]);v:=vector(3,[2,1,-4]);

> crossprod(u,v);crossprod(v,u);

The Standard Basis Vectors

> i:=vector(3,[1,0,0]);j:=vector(3,[0,1,0]);k:=vector(3,[0,0,1]);

> ixj:=crossprod(i,j);jxk:=crossprod(j,k);kxi:=crossprod(k,i);ixk:=crossprod(i,k);kxj:=crossprod(k,j);jxi:=crossprod(j,i);

Maple Syrup Proof of Lagrange's identity.

> left:=innerprod(crossprod(U,V),crossprod(U,V));

> right:=innerprod(U,U)*innerprod(V,V)-(innerprod(U,V))^2;

> left-right;

> simplify(left-right);

Other properties.

Challenge: Below are five Maple Syrup proofs of vector properties. Determine the properties being proved in each case.

one

> a:=crossprod(U,V);b:=crossprod(V,U);matadd(a,b);

two

> crossprod(scalarmul(U,c),V);crossprod(U,scalarmul(V,c));scalarmul(crossprod(U,V),c);

three

> V1:=crossprod(U,matadd(V,W));V2:=matadd(crossprod(U,V),crossprod(U,W));simplify(matadd(V1,-V2));

four

> V1:=crossprod(matadd(U,V),W);V2:=matadd(crossprod(U,W),crossprod(V,W));simplify(matadd(V1,-V2));

five

> V1:=innerprod(U,crossprod(V,W));V2:=innerprod(crossprod(U,V),W);V1-V2;

>

six

> V1:=crossprod(U,crossprod(V,W));V2:=matadd(scalarmul(V,innerprod(U,W)),scalarmul(W,-innerprod(U,V)));simplify(matadd(V1,-V2));

Triple Scalar Product: identify the identity that is established below.

> left:=innerprod(U,crossprod(V,W));Matrix:=stackmatrix(U,V,W);right:=det(Matrix);

> left-right;