What is the condition when general Cauchy inequality transforms to equality? Is there a general case when it happens?
I suspect you are talking about the Cauchy-Schwartz inequality, so I will answer assuming that my suspicion is correct. The inequality says that the inner product of two vectors in an inner product space is less than or equal to the product of the magnitude of the two vectors.
So, if you understand what inner product means, your question should answer itself. I will give you a hint: what happens if two non-zero vectors have an inner product of zero? What does that mean? That is essentially the opposite of when the Cauchy-Schwartz inequality will be an equality.