Two unit vectors u and v have dot product 1 2 . Find the length of their cross product.?

The dot product is 1/2 (not 12).

Let w be a unit vector in the plane P spanned by u and v, orthogonal

to u. Then v = au + bw. u.v = a = 1/2. a^2 + b^2 =1 ==> |b| = sqrt(3)/2

u x w = z, a unit vector orthogonal to both u and w

u x v = u x (au + bw) = b u x w = b z since u x u = 0

Therefore length of u x v = |b| = sqrt(3)/2

In general, for any two unit vectors u & v, u x v has length sqrt(1 - (u.v)^2 )

by the same reasoning.