Cross products and exact tests

Questions 1 to 5 of 8

1. 1. Take the points a = (1, 1), b = (3, 1) and c = (3, 4). What does the orientation test cross(b - a, c - a) give, and what kind of turn is it?

   Clockwise turn, cross product -6Left turn, cross product +6Straight line, cross product 0Left turn, cross product +2
2. 2. For a = (0, 0), b = (5, 1) and c = (2, 7), what is cross(b - a, c - a)? Give the integer.

   Type a number.

3. 3. An AI tool wrote the orientation test with int variables. The input coordinates go up to 10^9 in absolute value. Which statement is true?

   The sign of the cross product flips when the coordinates reach 10^9, so the turn is reported the wrong wayThe difference 2 * 10^9 does not fit in an int, so the subtraction wraps aroundThe difference 2 * 10^9 fits in an int, but the product of two such differences, up to about 4 * 10^18, does not, so the sign can be wrongAn int is safe, because a 32-bit product is reduced modulo the cross product before the sign is taken
4. 4. Which statements about the number types for the orientation test are true?

   Pick every answer that is true.

   long long holds values up to about 9.2 * 10^18For coordinates in [-10^9, 10^9], the largest cross product is about 8 * 10^18, which long long holds__int128 is part of the C++ standard, so every judge accepts itA double keeps about 16 decimal digits, so it can decide the sign of a cross product with coordinates up to 10^9
5. 5. A polar-angle sort used atan2 with if (fabs(a - b) < 1e-9) return false;. The input has u = (10^9, 10^9 - 1) and v = (10^9 - 1, 10^9 - 2), in that order. The output keeps u first. What is the right diagnosis?

   The tolerance treats two different directions as equal, so the order is wrong. The exact cross product is -1, so v is clockwise from u and must come firstatan2 returns NaN for coordinates this large, so the comparison is undefinedThe code is right, because a tolerance of 1e-9 is smaller than any real gap between anglesThe bug is the 32-bit int type, and the sort would be correct with long long