Exact Squared Distances, Signs, and Tie Groups
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261008-advance-20261010-r03-distance-formula-foundations · 2026-10-08 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Ordered Pair of Squared Euclidean Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[0, 0], [3, 4]] in order. Give their two squared L2 distances.
View answer and explanation
(0, 25)
Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (0, 25).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate
2. Ordered Pair of Squared Euclidean Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[1, -1], [2, 3]] in order. Give their two squared L2 distances.
View answer and explanation
(2, 13)
Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (2, 13).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate
3. Ordered Pair of Squared Euclidean Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[2, -1], [3, 4]] in order. Give their two squared L2 distances.
View answer and explanation
(5, 25)
Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (5, 25).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate
4. Ordered Pair of Squared Euclidean Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[3, -1], [4, 2]] in order. Give their two squared L2 distances.
View answer and explanation
(10, 20)
Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (10, 20).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate
5. Ordered Pair of Squared Euclidean Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[4, -1], [5, 3]] in order. Give their two squared L2 distances.
View answer and explanation
(17, 34)
Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (17, 34).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate
6. Ordered Pair of Squared Euclidean Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[5, -1], [6, 4]] in order. Give their two squared L2 distances.
View answer and explanation
(26, 52)
Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (26, 52).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate
7. Ordered Pair of Squared Euclidean Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[6, -1], [7, 2]] in order. Give their two squared L2 distances.
View answer and explanation
(37, 53)
Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (37, 53).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate
8. Ordered Pair of Squared Euclidean Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[7, -1], [8, 3]] in order. Give their two squared L2 distances.
View answer and explanation
(50, 73)
Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (50, 73).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate
9. Ordered Pair of Squared Euclidean Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[8, -1], [9, 4]] in order. Give their two squared L2 distances.
View answer and explanation
(65, 97)
Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (65, 97).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate
10. Ordered Pair of Squared Euclidean Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[9, -1], [10, 2]] in order. Give their two squared L2 distances.
View answer and explanation
(82, 104)
Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (82, 104).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate
11. Ordered Pair of Squared Euclidean Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[10, -1], [11, 3]] in order. Give their two squared L2 distances.
View answer and explanation
(101, 130)
Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (101, 130).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate
12. Ordered Pair of Squared Euclidean Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[11, -1], [12, 4]] in order. Give their two squared L2 distances.
View answer and explanation
(122, 160)
Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (122, 160).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate
13. Ordered Pair of Squared Euclidean Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[12, -1], [13, 2]] in order. Give their two squared L2 distances.
View answer and explanation
(145, 173)
Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (145, 173).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate
14. Ordered Pair of Squared Euclidean Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[13, -1], [14, 3]] in order. Give their two squared L2 distances.
View answer and explanation
(170, 205)
Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (170, 205).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate
15. Ordered Pair of Squared Euclidean Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[14, -1], [15, 4]] in order. Give their two squared L2 distances.
View answer and explanation
(197, 241)
Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (197, 241).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate
16. Ordered Pair of Squared Euclidean Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[15, -1], [16, 2]] in order. Give their two squared L2 distances.
View answer and explanation
(226, 260)
Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (226, 260).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate
17. Ordered Pair of Squared Euclidean Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[16, -1], [17, 3]] in order. Give their two squared L2 distances.
View answer and explanation
(257, 298)
Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (257, 298).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate
18. Ordered Pair of Squared Euclidean Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[17, -1], [18, 4]] in order. Give their two squared L2 distances.
View answer and explanation
(290, 340)
Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (290, 340).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate
19. Ordered Pair of Squared Euclidean Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[18, -1], [19, 2]] in order. Give their two squared L2 distances.
View answer and explanation
(325, 365)
Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (325, 365).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate
20. Ordered Pair of Squared Euclidean Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[19, -1], [20, 3]] in order. Give their two squared L2 distances.
View answer and explanation
(362, 409)
Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (362, 409).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate
21. Ordered Pair of Squared Euclidean Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[20, -1], [21, 4]] in order. Give their two squared L2 distances.
View answer and explanation
(401, 457)
Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (401, 457).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate
22. Ordered Pair of Squared Euclidean Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[21, -1], [22, 2]] in order. Give their two squared L2 distances.
View answer and explanation
(442, 488)
Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (442, 488).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate
23. Ordered Pair of Squared Euclidean Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[22, -1], [23, 3]] in order. Give their two squared L2 distances.
View answer and explanation
(485, 538)
Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (485, 538).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate
24. Ordered Pair of Squared Euclidean Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[23, -1], [24, 4]] in order. Give their two squared L2 distances.
View answer and explanation
(530, 592)
Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (530, 592).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate
25. Ordered Pair of Squared Euclidean Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[24, -1], [25, 2]] in order. Give their two squared L2 distances.
View answer and explanation
(577, 629)
Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (577, 629).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate
26. Candidate Sign and Squared-Distance Defect
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[3, 4], integer candidate c=-5. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.
View answer and explanation
(No, 0)
Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, 0).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate
27. Candidate Sign and Squared-Distance Defect
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[3, 4], integer candidate c=5. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.
View answer and explanation
(Yes, 0)
Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, 0).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate
28. Candidate Sign and Squared-Distance Defect
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[3, 4], integer candidate c=4. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.
View answer and explanation
(Yes, -9)
Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -9).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate
29. Candidate Sign and Squared-Distance Defect
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[0, 0], integer candidate c=0. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.
View answer and explanation
(Yes, 0)
Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, 0).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate
30. Candidate Sign and Squared-Distance Defect
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[4, 5], integer candidate c=7. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.
View answer and explanation
(Yes, 8)
Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, 8).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate
31. Candidate Sign and Squared-Distance Defect
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[5, 6], integer candidate c=-8. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.
View answer and explanation
(No, 3)
Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, 3).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate
32. Candidate Sign and Squared-Distance Defect
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[6, 7], integer candidate c=9. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.
View answer and explanation
(Yes, -4)
Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -4).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate
33. Candidate Sign and Squared-Distance Defect
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[7, 8], integer candidate c=-10. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.
View answer and explanation
(No, -13)
Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -13).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate
34. Candidate Sign and Squared-Distance Defect
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[8, 9], integer candidate c=11. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.
View answer and explanation
(Yes, -24)
Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -24).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate
35. Candidate Sign and Squared-Distance Defect
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[9, 10], integer candidate c=-12. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.
View answer and explanation
(No, -37)
Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -37).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate
36. Candidate Sign and Squared-Distance Defect
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[10, 11], integer candidate c=13. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.
View answer and explanation
(Yes, -52)
Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -52).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate
37. Candidate Sign and Squared-Distance Defect
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[11, 12], integer candidate c=-14. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.
View answer and explanation
(No, -69)
Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -69).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate
38. Candidate Sign and Squared-Distance Defect
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[12, 13], integer candidate c=15. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.
View answer and explanation
(Yes, -88)
Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -88).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate
39. Candidate Sign and Squared-Distance Defect
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[13, 14], integer candidate c=-16. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.
View answer and explanation
(No, -109)
Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -109).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate
40. Candidate Sign and Squared-Distance Defect
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[14, 15], integer candidate c=17. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.
View answer and explanation
(Yes, -132)
Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -132).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate
41. Candidate Sign and Squared-Distance Defect
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[15, 16], integer candidate c=-18. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.
View answer and explanation
(No, -157)
Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -157).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate
42. Candidate Sign and Squared-Distance Defect
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[16, 17], integer candidate c=19. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.
View answer and explanation
(Yes, -184)
Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -184).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate
43. Candidate Sign and Squared-Distance Defect
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[17, 18], integer candidate c=-20. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.
View answer and explanation
(No, -213)
Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -213).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate
44. Candidate Sign and Squared-Distance Defect
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[18, 19], integer candidate c=21. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.
View answer and explanation
(Yes, -244)
Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -244).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate
45. Candidate Sign and Squared-Distance Defect
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[19, 20], integer candidate c=-22. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.
View answer and explanation
(No, -277)
Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -277).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate
46. Candidate Sign and Squared-Distance Defect
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[20, 21], integer candidate c=23. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.
View answer and explanation
(Yes, -312)
Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -312).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate
47. Candidate Sign and Squared-Distance Defect
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[21, 22], integer candidate c=-24. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.
View answer and explanation
(No, -349)
Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -349).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate
48. Candidate Sign and Squared-Distance Defect
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[22, 23], integer candidate c=25. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.
View answer and explanation
(Yes, -388)
Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -388).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate
49. Candidate Sign and Squared-Distance Defect
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[23, 24], integer candidate c=-26. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.
View answer and explanation
(No, -429)
Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -429).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate
50. Candidate Sign and Squared-Distance Defect
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[24, 25], integer candidate c=27. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.
View answer and explanation
(Yes, -472)
Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -472).
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate
51. Distance Order with Explicit Tie Groups
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[0, 0], [0, 0], [0, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.
View answer and explanation
[[1, 2, 3]]
Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[1, 2, 3]].
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate
52. Distance Order with Explicit Tie Groups
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[1, 0], [0, -1], [2, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.
View answer and explanation
[[1, 2], [3]]
Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[1, 2], [3]].
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate
53. Distance Order with Explicit Tie Groups
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[2, 0], [1, 0], [1, 1]]. List index groups in increasing distance order, with indices increasing inside each tie group.
View answer and explanation
[[2], [3], [1]]
Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2], [3], [1]].
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate
54. Distance Order with Explicit Tie Groups
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[1, 1], [1, 0], [-1, 1]]. List index groups in increasing distance order, with indices increasing inside each tie group.
View answer and explanation
[[2], [1, 3]]
Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2], [1, 3]].
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate
55. Distance Order with Explicit Tie Groups
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[2, 1], [1, 2], [0, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.
View answer and explanation
[[3], [1, 2]]
Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[3], [1, 2]].
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate
56. Distance Order with Explicit Tie Groups
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[6, 0], [5, 1], [5, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.
View answer and explanation
[[3], [2], [1]]
Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[3], [2], [1]].
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate
57. Distance Order with Explicit Tie Groups
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[7, 1], [6, 0], [0, -6]]. List index groups in increasing distance order, with indices increasing inside each tie group.
View answer and explanation
[[2, 3], [1]]
Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2, 3], [1]].
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate
58. Distance Order with Explicit Tie Groups
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[7, 0], [8, 0], [9, 1]]. List index groups in increasing distance order, with indices increasing inside each tie group.
View answer and explanation
[[1], [2], [3]]
Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[1], [2], [3]].
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate
59. Distance Order with Explicit Tie Groups
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[9, 0], [8, 1], [8, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.
View answer and explanation
[[3], [2], [1]]
Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[3], [2], [1]].
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate
60. Distance Order with Explicit Tie Groups
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[10, 1], [9, 0], [0, -9]]. List index groups in increasing distance order, with indices increasing inside each tie group.
View answer and explanation
[[2, 3], [1]]
Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2, 3], [1]].
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate
61. Distance Order with Explicit Tie Groups
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[10, 0], [11, 0], [12, 1]]. List index groups in increasing distance order, with indices increasing inside each tie group.
View answer and explanation
[[1], [2], [3]]
Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[1], [2], [3]].
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate
62. Distance Order with Explicit Tie Groups
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[12, 0], [11, 1], [11, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.
View answer and explanation
[[3], [2], [1]]
Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[3], [2], [1]].
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate
63. Distance Order with Explicit Tie Groups
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[13, 1], [12, 0], [0, -12]]. List index groups in increasing distance order, with indices increasing inside each tie group.
View answer and explanation
[[2, 3], [1]]
Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2, 3], [1]].
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate
64. Distance Order with Explicit Tie Groups
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[13, 0], [14, 0], [15, 1]]. List index groups in increasing distance order, with indices increasing inside each tie group.
View answer and explanation
[[1], [2], [3]]
Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[1], [2], [3]].
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate
65. Distance Order with Explicit Tie Groups
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[15, 0], [14, 1], [14, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.
View answer and explanation
[[3], [2], [1]]
Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[3], [2], [1]].
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate
66. Distance Order with Explicit Tie Groups
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[16, 1], [15, 0], [0, -15]]. List index groups in increasing distance order, with indices increasing inside each tie group.
View answer and explanation
[[2, 3], [1]]
Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2, 3], [1]].
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate
67. Distance Order with Explicit Tie Groups
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[16, 0], [17, 0], [18, 1]]. List index groups in increasing distance order, with indices increasing inside each tie group.
View answer and explanation
[[1], [2], [3]]
Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[1], [2], [3]].
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate
68. Distance Order with Explicit Tie Groups
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[18, 0], [17, 1], [17, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.
View answer and explanation
[[3], [2], [1]]
Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[3], [2], [1]].
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate
69. Distance Order with Explicit Tie Groups
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[19, 1], [18, 0], [0, -18]]. List index groups in increasing distance order, with indices increasing inside each tie group.
View answer and explanation
[[2, 3], [1]]
Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2, 3], [1]].
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate
70. Distance Order with Explicit Tie Groups
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[19, 0], [20, 0], [21, 1]]. List index groups in increasing distance order, with indices increasing inside each tie group.
View answer and explanation
[[1], [2], [3]]
Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[1], [2], [3]].
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate
71. Distance Order with Explicit Tie Groups
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[21, 0], [20, 1], [20, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.
View answer and explanation
[[3], [2], [1]]
Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[3], [2], [1]].
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate
72. Distance Order with Explicit Tie Groups
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[22, 1], [21, 0], [0, -21]]. List index groups in increasing distance order, with indices increasing inside each tie group.
View answer and explanation
[[2, 3], [1]]
Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2, 3], [1]].
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate
73. Distance Order with Explicit Tie Groups
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[22, 0], [23, 0], [24, 1]]. List index groups in increasing distance order, with indices increasing inside each tie group.
View answer and explanation
[[1], [2], [3]]
Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[1], [2], [3]].
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate
74. Distance Order with Explicit Tie Groups
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[24, 0], [23, 1], [23, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.
View answer and explanation
[[3], [2], [1]]
Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[3], [2], [1]].
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate
75. Distance Order with Explicit Tie Groups
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[25, 1], [24, 0], [0, -24]]. List index groups in increasing distance order, with indices increasing inside each tie group.
View answer and explanation
[[2, 3], [1]]
Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2, 3], [1]].
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate
76. Sum of Two Integer Principal Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[0, 0], [0, 0]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.
View answer and explanation
0
Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 0.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate
77. Sum of Two Integer Principal Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[3, 4], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.
View answer and explanation
18
Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 18.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate
78. Sum of Two Integer Principal Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[0, 0], [3, 4]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.
View answer and explanation
5
Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 5.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate
79. Sum of Two Integer Principal Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[9, 12], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.
View answer and explanation
28
Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 28.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate
80. Sum of Two Integer Principal Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[12, 16], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.
View answer and explanation
33
Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 33.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate
81. Sum of Two Integer Principal Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[15, 20], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.
View answer and explanation
38
Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 38.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate
82. Sum of Two Integer Principal Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[18, 24], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.
View answer and explanation
43
Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 43.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate
83. Sum of Two Integer Principal Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[21, 28], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.
View answer and explanation
48
Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 48.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate
84. Sum of Two Integer Principal Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[24, 32], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.
View answer and explanation
53
Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 53.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate
85. Sum of Two Integer Principal Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[27, 36], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.
View answer and explanation
58
Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 58.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate
86. Sum of Two Integer Principal Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[30, 40], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.
View answer and explanation
63
Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 63.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate
87. Sum of Two Integer Principal Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[33, 44], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.
View answer and explanation
68
Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 68.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate
88. Sum of Two Integer Principal Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[36, 48], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.
View answer and explanation
73
Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 73.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate
89. Sum of Two Integer Principal Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[39, 52], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.
View answer and explanation
78
Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 78.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate
90. Sum of Two Integer Principal Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[42, 56], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.
View answer and explanation
83
Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 83.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate
91. Sum of Two Integer Principal Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[45, 60], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.
View answer and explanation
88
Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 88.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate
92. Sum of Two Integer Principal Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[48, 64], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.
View answer and explanation
93
Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 93.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate
93. Sum of Two Integer Principal Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[51, 68], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.
View answer and explanation
98
Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 98.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate
94. Sum of Two Integer Principal Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[54, 72], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.
View answer and explanation
103
Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 103.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate
95. Sum of Two Integer Principal Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[57, 76], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.
View answer and explanation
108
Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 108.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate
96. Sum of Two Integer Principal Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[60, 80], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.
View answer and explanation
113
Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 113.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate
97. Sum of Two Integer Principal Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[63, 84], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.
View answer and explanation
118
Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 118.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate
98. Sum of Two Integer Principal Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[66, 88], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.
View answer and explanation
123
Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 123.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate
99. Sum of Two Integer Principal Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[69, 92], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.
View answer and explanation
128
Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 128.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate
100. Sum of Two Integer Principal Distances
In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[72, 96], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.
View answer and explanation
133
Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 133.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate
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