Classification of vector sequences
As you can't disclose much detail, I'm forced to be a bit generic in my answer. I hope it will be helpful nevertheless. First of all, I would only consider reducing the sequences before classification (be it by using the dot product or something else) if you can make sure that you don't lose information you need for classification afterwards. So this approach is only feasible if you have some insight into the nature of the classification. To give a simple example: if the class label is just the number of vectors in your sequence, you won't be very successfull in predicting the class label from the dot product.

Hence, I would take the full sequence as an input for classification, and impose a maximum on the sequence length you want to consider. You might do this by first finding the maximum sequence length m in your training set and then turning each sequence of 50-dimensional vectors into one vector of dimension 50*m, possibly with some missing values at the end if your sequence doesn't have maximum length. You will probably want to get rid of these missing values and you may want to simply replace them by zeros. 

There's two roads you can go from here: 
1.) You directly apply classification methods known to be suitable for high dimensions. Try something simple that doesn't need much tuning like naive Bayes. This way you can see whether this approach is feasible without losing too much time if it isn't.
2.) You try first to reduce the dimension and understand the nature of the classification better. You may want to use something like principal components analysis or analyse correlation/association between each vector component and the class label. If you're successful, you know how to properly reduce the dimension of your input before applying classification.

If you would like to follow any of these ideas, please keep in mind that the concrete details of your data and the classification may render any of the ideas proposed above infeasible. So please be careful to check against any details you know but can't post here before trying to make sure you're not wasting your time.