Tensor property

[typhoonss821]

1. The problem statement, all variables and given/known data

Let A_{i} , B_{j} , C_{k}be vectors.
Is A_{1} B_{i} + A_{2} B_{j} + A_{3} B_{k} a tensor?

2. Relevant equations
T'_{ijk} = R_{il} R_{jm} R_{kn} T_{ijk}


3. The attempt at a solution
If T is a tensor of rank 3, then T'_{ijk} = R_{il} R_{jm} R_{kn} T_{ijk} where R^\intercal R =I . But I am not sure how to use this relation in the problem. > <

I think it is to use the property of vector A'_{i} = R_{il} A_{i} , but difficult appears when I try this way.

Please help...