On approximately what date will a line drawn between the last 2 stars in the cup of the big dipper point most directly at polaris?

The  two last stars of the Dipper are the Pointers; a line through them passes close to the North Star. Imagine that the North Star is the center of the clock, and an hour hand passes up to the Pointer stars. But it is a 24 hour clock, and the hour hand change backwards! Midnight = Zero hours = twenty-four hours is at the top, 6 hours = 6 AM is on the left, 12 hours = noon is at the bottom, and 18 hours = 6 PM is on the right. The clock above reads 23 hours. On the night of March 6, you can read the time directly off the clock. Daylight Saving Time = Dipper Clock Reading - 2 X (number of months after March 6) + 1 hour. 

see the picture for the vector

the magnitude of the vector by pythagorean theorem

r = √( (-10)^2 + (3)^2 ) = √109

magnitude = √109 units

the direction angle between vector and x-axis is

tan^(-1)(3/10) = 16.699

so the direction is approximately 16.7 degrees north of west