Is this condition correct for right hand thumb rule? If the three vectors A,B and C satisfy the relation A⋅B=0 and A⋅C=0, then vector A is parallel to :
correct answer is : B x C
Below , I have made is the diagram for the above Q. There are two possible condition I see for N vector here.

*

*When both C and B vector are present at Y axis and are perpendicular to A vector which is at X axis.


*One of the vectors either C or B(In my diagram , I chose B) is present at Z axis.
Form : 1) A.B=0 , A.C=0. Conditions are met. Now , to check if B x C is parallel to A vector. I get direction for B x C perpendicular to the plane using right hand thumb rule . Therefore , it is not parallel to A which is a wrong answer.


*Here , A.B=0 , A.C=0 & also , B.C=0. When I use right hand thumb rule here , I get B x C parallel to vector A.

My Q here is that :

*

*Can we say this is a condition for right hand thumb rule where angle between two vectors is less than 180. That’s why my 1st case isn’t working.


Edit: Right hand thumb rule :

 A: I think the question is implying that two vectors being anti-parallel is regarded as being parallel.
In your first scenario, since $B$ and $C$ are anti-parallel to each other, $B\times C=\vec{0}$. Since all vectors are parallel to the zero vector, we have that $A$ is parallel to $B\times C$.
As for the original problem, since $A\cdot B=A\cdot C=0$, we get that $A$ is perpendicular to $B$ and $C$. Therefore, $A$ should be parallel to the vector perpendicular to $B$ and $C$, which is $B\times C$.
A: Try to use the rule. It says to curl the fingers around the smaller angle between the vectors. The two angles here are the same, $180^\circ$. So, you can curl your fingers in two directions both of which are valid, and give different directions. With an angle less than $180^\circ$, this ambiguity wouldn't have arisen.
Why does this happen? Because the magnitude of the product $bc\sin\theta$ becomes $0$ when $\theta=180^\circ$ implying overall $\mathbf{b}\times\mathbf{c}=\mathbf{0}$ and the null vector has an arbitrary direction, hence the rule giving you two directions.
And as for the problem, we consider $\mathbf{0}$ to be parallel, perpendicular or antiparallel to every vector because it has an arbitrary direction.
Hope this was helpful. Ask anything if not clear :)
