Question

how can (-5, -4) compare to (2, 3)

Answer 1

There are several meaningful ways to compare the points (-5, -4) and (2, 3). Here are common comparisons with results:

- Quadrant and sign:
- (-5, -4) is in quadrant III (both coordinates negative).
- (2, 3) is in quadrant I (both coordinates positive).

- Coordinate-wise:
- -5 < 2 and -4 < 3, so (-5, -4) is smaller in both x and y.

- Distance from the origin (Euclidean norm):
- |(-5, -4)| = sqrt(5^2 + 4^2) = sqrt(41) ≈ 6.403
- |(2, 3)| = sqrt(2^2 + 3^2) = sqrt(13) ≈ 3.606
- So (-5, -4) is farther from the origin.

- Manhattan (L1) norm:
- 9 vs 5, so (-5, -4) has the larger L1 distance.

- Distance between the two points:
- d = sqrt((2 - (-5))^2 + (3 - (-4))^2) = sqrt(7^2 + 7^2) = sqrt(98) = 7√2 ≈ 9.899.

- Midpoint:
- ((-5+2)/2, (-4+3)/2) = (-1.5, -0.5).

- As vectors (dot product and angle):
- (-5,-4) · (2,3) = -10 - 12 = -22 (negative).
- Angle between them cosθ = -22 / (|(-5,-4)|·|(2,3)|) ≈ -0.95 → θ ≈ 162° (they point roughly opposite directions).

- Lexicographic order (compare x first): (-5, -4) < (2, 3).

If you meant a specific type of comparison (e.g., ordering, distance, angles, projections), tell me which and I can expand on that.