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Work + Unit Vectors + Dot Product
(Each direction is a vector)
Let 
represent the direction horizontally north, 
represent northeast (halfway between north and east), 
represent vertically upward, and so on. Each direction specification can be thought of as a unit vector. Rank from the largest to the smallest the following dot products. Note that zero is larger than a negative number. If two quantities are equal, display that fact in your ranking.
a) N*N = N*UP > N*NE = UP*DOWN > N*E > N*S > E*E = SE*S
b) N*N = E*E > N*NE = SE*S > N*E = N*UP > N*S = UP*DOWN
c) N*N > E*E = N*NE > SE*S = N*E > N*UP > N*S = UP*DOWN
d) N*N = E*E > N*NE = SE*S > N*S = UP*DOWN > N*E = N*UP
e) N*N = N*UP > N*NE > UP*DOWN > N*E > N*SE > E*E = SE*S
represent the direction horizontally north, represent northeast (halfway between north and east), represent vertically upward, and so on. Each direction specification can be thought of as a unit vector. Rank from the largest to the smallest the following dot products. Note that zero is larger than a negative number. If two quantities are equal, display that fact in your ranking.
a) N*N = N*UP > N*NE = UP*DOWN > N*E > N*S > E*E = SE*S
b) N*N = E*E > N*NE = SE*S > N*E = N*UP > N*S = UP*DOWN
c) N*N > E*E = N*NE > SE*S = N*E > N*UP > N*S = UP*DOWN
d) N*N = E*E > N*NE = SE*S > N*S = UP*DOWN > N*E = N*UP
e) N*N = N*UP > N*NE > UP*DOWN > N*E > N*SE > E*E = SE*S
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