Exercise 1: Facing Angle and Turn Direction for a New Sprite — Possible Solution
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GIVEN
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f = [0, 1]   (facing "up")
d = [-2, 2]  (direction to the player)

STEP 1: THE DOT PRODUCT AND ANGLE (STEP 1's TECHNIQUE)
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f . d = (0)(-2) + (1)(2) = 0 + 2 = 2

|f| = sqrt(0^2 + 1^2) = 1
|d| = sqrt((-2)^2 + 2^2) = sqrt(4 + 4) = sqrt(8) ~= 2.828

cos(theta) = (f . d) / (|f| |d|) = 2 / (1 x 2.828) ~= 0.7071

theta = arccos(0.7071) = 45 degrees

The player is 45 degrees off of directly ahead - well within a
typical detection cone, similar in spirit to Step 1's own ~53.1
degree result.

STEP 2: THE 2D CROSS PRODUCT AND TURN DIRECTION (STEP 2's TECHNIQUE)
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f1 d2 - f2 d1 = (0)(2) - (1)(-2) = 0 + 2 = 2

The result is positive, meaning (per this chapter's own Step 2
convention) d is counter-clockwise from f - the player is to the
sprite's LEFT. The AI should turn left.

WHY THIS WORKS AS AN ANSWER
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Both calculations reuse this chapter's own Step 1 and Step 2
techniques exactly - the dot product for the angle, the 2D cross
product's sign for the turn direction - applied to the new f and d
values rather than simply re-describing the original worked example.