Suppose for the sake of argument that our database objects are two-dimensional vectors. There are four operations we can perform on vectors, and each will have its own type of lock.

• change the value along the X-axis (X-lock)
• change the value along the Y-axis (Y-lock)
• change the angle of the vector (A-lock)
• change the magnitude of the vector (M-lock) Answer the following questions:
• Which pairs of operations commute? For example, if we rotate the vector so its angle is 120 degrees and then change the x-coordinate to be 10, is that the same as first changing the x-coordinate to 10 and then changing the angle to 120 degrees?
• Based on your last answer, what is the compatibility matrix for the four types of locks?
• Suppose we changed the four operations so that instead of giving new values for a measure, the operations incremented the measure (e.g., add 10 to the x-coordinate or rotate the vector 45 degrees anti-clockwise). What would the compatibility matrix then be?