We are given that ||e|| = 1, ||f|| = 1.
Since ||e + f|| = sqrt(3/2), we have
3/2 = (e + f) dot (e + f)
= (e dot e) + 2(e dot f) + (f dot f)
= ||e||^2 + 2(e dot f) + ||f||^2
= 1^2 + 2(e dot f) + 1^2
= 2 + 2(e dot f).
So e dot f = -1/4.
Therefore,
||2e - 3f||^2 = (2e - 3f) dot (2e - 3f)
= 4(e dot e) - 12(e dot f) + 9(f dot f)
= 4||e||^2 - 12(e dot f) + 9||f||^2
= 4(1)^2 - 12(-1/4) + 9(1)^2
= 4 + 3 + 9
= 16.
