Question

Show that R^-1(a) = R(-a). This equation shows that a rotation through a negative angle is equivalent to an inverse transformation.

 

 

Answer 1

The rotation matrix R(a) is defined as follows,

 

Here we need to show that R-1(a) = R(-a). For solving this, we will define R(a) by declaring “a” as a symbolic variable. We will find R-1(a) and R(-a) and then show that both are same. The MATLAB code for it are as follows,

 

Input:

syms a %defining symbolic variable

R=[cos(a),sin(a);-sin(a),cos(a)]; %Defining rotation matrix

Rinv=simplify(inv(R)); %finding the inverse of R

Ra=simplify(subs(R,a,-a)); %finding the value of R(-a)

disp(\'Inverse of Rotation matrix is\')

disp(Rinv)

disp(\'Matrix R(-a) is\')

disp(Ra)

disp(\'\')

disp(\'Hence we proved that R(-a) is equal to inverse of Rotation matrix\')

 

Output:

 

Hence proved that R-1(a) = R(-a).

Hence proved that R-1(a) = R(-a).