Your statement of the contrapositive is incorrect. The only way to state the contrapositive is to have the "not" affect the entire "if" part. The correct contrapositive statement would be the following:
Let . If , then it is not the case that for every , we have .
The way you worded the contrapositive, the "not" applied only to the inequality of the dot product, and it did not apply to the quantifier "for all". Do you see what I mean?