"What's our vector, Victor?": What does this mean?
A vector is defined by a direction and magnitude.  In aviation these represent your heading (the direction) and your speed (the magnitude).  However, in normal aviation usage "vector" only refers to the heading and other nomenclature is used to assign/report speeds.

Example:

"Jetlink 1234, Cleveland Center, turn left heading 250 for traffic"

In which case the vector assigned by ATC is 250 degrees, or 20 degrees south of magnetic west.

To answer the question "What is our vector, Victor?", Victor should state the heading they have been assigned to fly from ATC.  

An alternative question would be "What is your heading?" and then you would answer with the direction you are currently flying.