Derive relations for components by rotation of axes

[briteliner]

Homework Statement

x1,x2) are the components of a 2 dimensional vector r when referred to cartesian axes along the directions i,j. derive the relations
x1'= cosΘ x1+sinΘ x2
x2'=-sinΘ x1+cosΘ x2
for the components (x1',x2') or r referred to new axes i',j' obtained by a rotation of the axes through an angle Θ about the k direction

Homework Equations

The Attempt at a Solution

i just wrote that r=x^2+y^2 which in this case would be x1^2+x2^2 and then accounted for rotation by multiplying by sin or cos theta and proving that x1^2+x2^2=x1'^2+x2'^2 but it don't think its sufficient

 

[Dick]

What's (1,0) rotated by an angle theta? Ditto for (0,1). To get those just draw a right triangle in the xy plane with angle theta at the origin. Use trig. Then (x1,x2)=x1*(1,0)+x2*(0,1).