Given the Rotation Matrix (M), rotate any 2D vector in a counterclockwise by theta (x), then show that M'x rotates the vector clockwise back to the original position

I have a homework assignment, and I am asked to rotate a vector (v) by the rotation matrix (M), by any angle (x) in a counter clockwise direction, then show that M'x rotates the vector back to the original position.

My code looks like this:

%theta=x

%R=M

v=[100 100]

x=45

R=[cosd(x) -sind(x);sind(x) cosd(x)]

vR=v*R

quiver(0,0,v(1),v(2))

hold on

quiver(0,0,vR(1),vR(2))

S=R'

vS=v*S

quiver(0,0,v(1),v(2))

hold on

quiver(0,0,vS(1),vS(2))

My plot looks like this:

[Walter Roberson]

Okay but what is your question? And where is your proof that was required? A plot is not proof.

[Jimmy Cleveland]

Walter, Thanks for the reply. Not sure how I would go about proving that the transpose of the rotation matrix would rotate the vector back to the original position. Thanks, Jimmy

[Walter Roberson]

Take the formula you have for R and write out the terms of

 R.' * R

and remember one of the fundamental properties relating sine and cosine.

I recommend you work in radians to make the formulas simpler

Answer 1

[Jimmy Cleveland]

Thanks to both of you!

Answer 2

[Image Analyst]

Try using ismembertol() to determine if v and vs are within some small number, like 0.00001 or something. Then, you got the rotation matrix on the wrong side. Try it this way.

v = [100; 100] % Initialize to point along the 45 degree line.
x = 45; % Define angle to rotate by.
% Construct rotation matrix.
M = [cosd(x) -sind(x); sind(x) cosd(x)]
vR = M * v

% Plot original vector.
subplot(1, 2, 1); % Plot in the left plot.
quiver(0, 0, v(1), v(2), 'LineWidth', 2)
text(v(1), v(2), 'Original');
grid on;
hold on
axis equal;
xlim([-5,150]);
ylim([-5,150]);
% Plot rotated vector.
quiver(0, 0, vR(1), vR(2), 'LineWidth', 2)
text(vR(1), vR(2), 'Rotated');

% Rotate vR back to v location.
vS = M' * vR
% Plot rotated vector.
subplot(1, 2, 2); % Plot in the left plot.
quiver(0, 0, vR(1), vR(2), 'LineWidth', 2)
text(vR(1), vR(2), 'Rotated');
grid on;
hold on
axis equal;
xlim([-5,150]);
ylim([-5,150]);

% Plot restored vector.
quiver(0, 0, vS(1), vS(2), 'LineWidth', 2)
text(vS(1), vS(2), 'Rotated back');

theyMatch = ismembertol(v, vS, 0.0001)
if all(theyMatch)
  uiwait(helpdlg('They match'));
else
  uiwait(helpdlg('They do not match'));
end

As for why quiver does not go all the way out to the point it should, like (100,100), I have no idea.