matlabrotationlinear-algebrageometry-surfacecross-product

# Rotate a vector by 90 degree via cross product

I am trying to understand a code wherein the authors have rotated vectors in 3D by 90 degrees. They used a cross-product to do so. How cross-product can incorporate rotation by 90 degrees? Any explanation would be helpful.

``````    % Inputs:
%   V  #vertices by dim list of mesh vertex positions
%   F  #faces by simplex-size list of mesh face indices
% Gradient of a scalar function defined on piecewise linear elements (mesh)
% is constant on each triangle i,j,k:
% grad(Xijk) = (Xj-Xi) * (Vi - Vk)^R90 / 2A + (Xk-Xi) * (Vj - Vi)^R90 / 2A
% grad(Xijk) = Xj * (Vi - Vk)^R90 / 2A + Xk * (Vj - Vi)^R90 / 2A +
%             -Xi * (Vi - Vk)^R90 / 2A - Xi * (Vj - Vi)^R90 / 2A
% where Xi is the scalar value at vertex i, Vi is the 3D position of vertex
% i, and A is the area of triangle (i,j,k). ^R90 represent a rotation of
% 90 degrees
%
% renaming indices of vertices of triangles for convenience
i1 = F(:,1); i2 = F(:,2); i3 = F(:,3);
% #F x 3 matrices of triangle edge vectors, named after opposite vertices
v32 = V(i3,:) - V(i2,:);  v13 = V(i1,:) - V(i3,:); v21 = V(i2,:) - V(i1,:);

% area of parallelogram is twice area of triangle
% area of parallelogram is || v1 x v2 ||
n  = cross(v32,v13,2);

% This does correct l2 norm of rows so that it contains #F list of twice
% triangle areas
dblA = normrow(n);

% now normalize normals to get unit normals
u = normalizerow(n);

% rotate each vector 90 degrees around normal ---- ????
eperp21 = bsxfun(@times,cross(u,v21),1./dblA);
eperp13 = bsxfun(@times,cross(u,v13),1./dblA);
``````

Solution

• In 3d the cross product `a x b` of two vectors `a` and `b` results in a vector `p := a x b` that is perpendicular to both `a` and `b`.

This means if you cross-multiply a vector with an unit vector `u` that represents the rotation axis, you will get a vector that is rotated 90 degrees around the rotation axis.