I post the question with hints.

9. Prove that a nonzero tangent vector v = vix, + v2x, is a principal vector if and only if 2 Vi -V,V2 F E G = 0. L M N Hint: v is principal if and only if S(v) X v = 0. Expand S(v) x v. This vector is zero if and only if its dot product with X X X, is zero. Use the Lagrange identity For vectors x, y, v, w in R}, the Lagrange identity X. V X. W (x * y)(vxw) = y. V y.w 9. Prove that a nonzero tangent vector v = vix, + v2x, is a principal vector if and only if 2 Vi -V,V2 F E G = 0. L M N Hint: v is principal if and only if S(v) X v = 0. Expand S(v) x v. This vector is zero if and only if its dot product with X X X, is zero. Use the Lagrange identity For vectors x, y, v, w in R}, the Lagrange identity X. V X. W (x * y)(vxw) = y. V y.w