Question

if the sum of the magnitudes of both vectors is 10 and the magnitude of resultant is 12 and it is perpendicular to the shortest vector then find the magnitudes of both

Answer 1

Assuming X as shorter vector

R^2=Y^2-X^2
144=(Y+X)(Y-X)=10(Y-X)

Now solve for Y-X

Then you can use values for Y-X and Y+X to find X and Y.