Let N cap represent the direction horizontally north, NE cap represent northeast (halfway between north and east), 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 cap middot N cap (b) N cap middot NE cap (c) N cap middot S cap (d) N cap middot E cap (e) SE cap middot S cap

The cardinal points can in fact be worked as a Cartesian coordinate system (xy), so the relationships for the scalar product (dot product) are valid for both

A . B = A B cos θ

Where the bold in A and B indicate vectors, the values on the right are the modules of (A, B) and θ is the angle between the two vectors.

Let's make the products that are requested.

a) N. N the angle between the two vectors is zero and cos 0 = 1, the result the product squared

N.N = N N cos 0 º

N.N = N² 1

If N is a unit vector, module equal to one (1), N² = N = 1

N.N = 1

b) N. NE. The North-East are 45º with respect to the north, so the angle between the two vectors is 45º

N.NE = N Ne cos 45

N. NE = N Ne 0.707

If N and Ne are unitary

N. NE = 0.707

c) N. S these two vectors are on the same vertical line one up (N) and gold down (S), so the angle is 180º

N.S = N S cos 180

N.S = -1

d) N.E one goes up and the other to the right, the angle between the 90º doses

N.E = N E cos 90

N.E = 0

e) SE.S one of these vectors is 45º in the fourth quadrant (where x is positive and y is negative) and the other points vertically downwards (S) the angle between the two is 45º

In general, for these products the angle is measured from the positive part of the x-axis (EAST), therefore, 270º must be added to the Angle found

SE. S = Se S cos (270 + 45)

SE.S = 0.707

Order products from highest to lowest

1 N.N

2 N.Ne

SE.S

3 N.E

4 N.S

The N.NE and SE products. has the same numerical value, but one is in the first quadrant and the other in fourth quadrant