Is this condition correct for right hand thumb rule

cross productvectors

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.

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

  2. 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.

  1. 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 :

  1. 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.

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Edit: Right hand thumb rule :

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Best Answer

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 :)

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