For what values of b are the vectors ⟨−110,b,10⟩ and ⟨b, b2,b⟩ orthogonal?

Two vectors are orthogonal if their dot product is zero. The dot product is the sum of the multiplications of entries with same index:

.

So, in your case,

So, the two vectors are orthogonal if and only if

The solution is indeed trivial: in that case, the second vector is the null vector, which is orthogonal to every possible vector.