My book gives two contrasting approaches to solving the same type of problems.
Problem 1
If the vectors $\vec{c}$, $\vec{a}=xi+yj+zk$, and $\vec{b}=j$ are such that $\vec{a}$, $\vec{c}$, and $\vec{b}$ form a right-handed system, find $\vec{c}$. (Subjective type)
Book approach: they've assumed $\vec{c}=\vec{b}\times\vec{a}$ and solved for $\vec{c}$.
Problem 2
What is the condition for $\vec{a}$, $\vec{b}$ and $\vec{c}$ to form a right-handed system? (Fill in the blanks type)
Book approach: Put $[\vec{a}\text{ }\vec{b}\text{ }\vec{c}]>0$
As you can see, there's no relation between the two approaches. I personally am favoring the second method because its result follows from the definition of the scalar triple product itself (volume of parallelopiped of vectors taken in order...).
My question: Which approach is correct?