Ordered L-M-T Dimension Vectors and Necessary Consistency
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batch-20261013-advance-20261020-r05-dimensional-analysis-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Product Dimension by Componentwise Exponent Addition
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-3, -1, -3], right=[-3, -2, -2]. Give (left, right, product exponent vector).
View answer and explanation
([-3, -1, -3], [-3, -2, -2], [-6, -3, -5])
Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, -1, -3], [-3, -2, -2], [-6, -3, -5]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-product · candidate
2. Product Dimension by Componentwise Exponent Addition
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-2, -1, -1], right=[0, -1, -2]. Give (left, right, product exponent vector).
View answer and explanation
([-2, -1, -1], [0, -1, -2], [-2, -2, -3])
Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, -1, -1], [0, -1, -2], [-2, -2, -3]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-product · candidate
3. Product Dimension by Componentwise Exponent Addition
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-1, -1, 1], right=[3, 0, -2]. Give (left, right, product exponent vector).
View answer and explanation
([-1, -1, 1], [3, 0, -2], [2, -1, -1])
Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, -1, 1], [3, 0, -2], [2, -1, -1]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-product · candidate
4. Product Dimension by Componentwise Exponent Addition
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[0, -1, 3], right=[-1, 1, -2]. Give (left, right, product exponent vector).
View answer and explanation
([0, -1, 3], [-1, 1, -2], [-1, 0, 1])
Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, -1, 3], [-1, 1, -2], [-1, 0, 1]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-product · candidate
5. Product Dimension by Componentwise Exponent Addition
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[1, -1, -2], right=[2, 2, -2]. Give (left, right, product exponent vector).
View answer and explanation
([1, -1, -2], [2, 2, -2], [3, 1, -4])
Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, -1, -2], [2, 2, -2], [3, 1, -4]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-product · candidate
6. Product Dimension by Componentwise Exponent Addition
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[2, -1, 0], right=[-2, -2, -1]. Give (left, right, product exponent vector).
View answer and explanation
([2, -1, 0], [-2, -2, -1], [0, -3, -1])
Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, -1, 0], [-2, -2, -1], [0, -3, -1]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-product · candidate
7. Product Dimension by Componentwise Exponent Addition
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[3, -1, 2], right=[1, -1, -1]. Give (left, right, product exponent vector).
View answer and explanation
([3, -1, 2], [1, -1, -1], [4, -2, 1])
Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, -1, 2], [1, -1, -1], [4, -2, 1]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-product · candidate
8. Product Dimension by Componentwise Exponent Addition
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-3, 0, -3], right=[-3, 0, -1]. Give (left, right, product exponent vector).
View answer and explanation
([-3, 0, -3], [-3, 0, -1], [-6, 0, -4])
Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 0, -3], [-3, 0, -1], [-6, 0, -4]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-product · candidate
9. Product Dimension by Componentwise Exponent Addition
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-2, 0, -1], right=[0, 1, -1]. Give (left, right, product exponent vector).
View answer and explanation
([-2, 0, -1], [0, 1, -1], [-2, 1, -2])
Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 0, -1], [0, 1, -1], [-2, 1, -2]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-product · candidate
10. Product Dimension by Componentwise Exponent Addition
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-1, 0, 1], right=[3, 2, -1]. Give (left, right, product exponent vector).
View answer and explanation
([-1, 0, 1], [3, 2, -1], [2, 2, 0])
Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 0, 1], [3, 2, -1], [2, 2, 0]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-product · candidate
11. Product Dimension by Componentwise Exponent Addition
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[0, 0, 3], right=[-1, -2, 0]. Give (left, right, product exponent vector).
View answer and explanation
([0, 0, 3], [-1, -2, 0], [-1, -2, 3])
Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 0, 3], [-1, -2, 0], [-1, -2, 3]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-product · candidate
12. Product Dimension by Componentwise Exponent Addition
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[1, 0, -2], right=[2, -1, 0]. Give (left, right, product exponent vector).
View answer and explanation
([1, 0, -2], [2, -1, 0], [3, -1, -2])
Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 0, -2], [2, -1, 0], [3, -1, -2]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-product · candidate
13. Product Dimension by Componentwise Exponent Addition
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[2, 0, 0], right=[-2, 0, 0]. Give (left, right, product exponent vector).
View answer and explanation
([2, 0, 0], [-2, 0, 0], [0, 0, 0])
Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 0, 0], [-2, 0, 0], [0, 0, 0]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-product · candidate
14. Product Dimension by Componentwise Exponent Addition
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[3, 0, 2], right=[1, 1, 0]. Give (left, right, product exponent vector).
View answer and explanation
([3, 0, 2], [1, 1, 0], [4, 1, 2])
Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 0, 2], [1, 1, 0], [4, 1, 2]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-product · candidate
15. Product Dimension by Componentwise Exponent Addition
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-3, 1, -3], right=[-3, 2, 0]. Give (left, right, product exponent vector).
View answer and explanation
([-3, 1, -3], [-3, 2, 0], [-6, 3, -3])
Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 1, -3], [-3, 2, 0], [-6, 3, -3]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-product · candidate
16. Product Dimension by Componentwise Exponent Addition
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-2, 1, -1], right=[0, -2, 1]. Give (left, right, product exponent vector).
View answer and explanation
([-2, 1, -1], [0, -2, 1], [-2, -1, 0])
Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 1, -1], [0, -2, 1], [-2, -1, 0]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-product · candidate
17. Product Dimension by Componentwise Exponent Addition
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-1, 1, 1], right=[3, -1, 1]. Give (left, right, product exponent vector).
View answer and explanation
([-1, 1, 1], [3, -1, 1], [2, 0, 2])
Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 1, 1], [3, -1, 1], [2, 0, 2]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-product · candidate
18. Product Dimension by Componentwise Exponent Addition
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[0, 1, 3], right=[-1, 0, 1]. Give (left, right, product exponent vector).
View answer and explanation
([0, 1, 3], [-1, 0, 1], [-1, 1, 4])
Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 1, 3], [-1, 0, 1], [-1, 1, 4]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-product · candidate
19. Product Dimension by Componentwise Exponent Addition
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[1, 1, -2], right=[2, 1, 1]. Give (left, right, product exponent vector).
View answer and explanation
([1, 1, -2], [2, 1, 1], [3, 2, -1])
Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 1, -2], [2, 1, 1], [3, 2, -1]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-product · candidate
20. Product Dimension by Componentwise Exponent Addition
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[2, 1, 0], right=[-2, 2, 1]. Give (left, right, product exponent vector).
View answer and explanation
([2, 1, 0], [-2, 2, 1], [0, 3, 1])
Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 1, 0], [-2, 2, 1], [0, 3, 1]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-product · candidate
21. Product Dimension by Componentwise Exponent Addition
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[3, 1, 2], right=[1, -2, 2]. Give (left, right, product exponent vector).
View answer and explanation
([3, 1, 2], [1, -2, 2], [4, -1, 4])
Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 1, 2], [1, -2, 2], [4, -1, 4]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-product · candidate
22. Product Dimension by Componentwise Exponent Addition
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-3, 2, -3], right=[-3, -1, 2]. Give (left, right, product exponent vector).
View answer and explanation
([-3, 2, -3], [-3, -1, 2], [-6, 1, -1])
Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 2, -3], [-3, -1, 2], [-6, 1, -1]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-product · candidate
23. Product Dimension by Componentwise Exponent Addition
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-2, 2, -1], right=[0, 0, 2]. Give (left, right, product exponent vector).
View answer and explanation
([-2, 2, -1], [0, 0, 2], [-2, 2, 1])
Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 2, -1], [0, 0, 2], [-2, 2, 1]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-product · candidate
24. Product Dimension by Componentwise Exponent Addition
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-1, 2, 1], right=[3, 1, 2]. Give (left, right, product exponent vector).
View answer and explanation
([-1, 2, 1], [3, 1, 2], [2, 3, 3])
Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 2, 1], [3, 1, 2], [2, 3, 3]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-product · candidate
25. Product Dimension by Componentwise Exponent Addition
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[0, 2, 3], right=[-1, 2, 2]. Give (left, right, product exponent vector).
View answer and explanation
([0, 2, 3], [-1, 2, 2], [-1, 4, 5])
Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 2, 3], [-1, 2, 2], [-1, 4, 5]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-product · candidate
26. Quotient Dimension with Ordered Exponent Subtraction
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[-3, -1, -3], denominator=[-3, -2, -2]. Give (numerator vector, denominator vector, quotient exponent vector).
View answer and explanation
([-3, -1, -3], [-3, -2, -2], [0, 1, -1])
Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, -1, -3], [-3, -2, -2], [0, 1, -1]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-quotient · candidate
27. Quotient Dimension with Ordered Exponent Subtraction
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[-2, -1, -1], denominator=[0, -1, -2]. Give (numerator vector, denominator vector, quotient exponent vector).
View answer and explanation
([-2, -1, -1], [0, -1, -2], [-2, 0, 1])
Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, -1, -1], [0, -1, -2], [-2, 0, 1]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-quotient · candidate
28. Quotient Dimension with Ordered Exponent Subtraction
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[-1, -1, 1], denominator=[3, 0, -2]. Give (numerator vector, denominator vector, quotient exponent vector).
View answer and explanation
([-1, -1, 1], [3, 0, -2], [-4, -1, 3])
Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, -1, 1], [3, 0, -2], [-4, -1, 3]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-quotient · candidate
29. Quotient Dimension with Ordered Exponent Subtraction
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[0, -1, 3], denominator=[-1, 1, -2]. Give (numerator vector, denominator vector, quotient exponent vector).
View answer and explanation
([0, -1, 3], [-1, 1, -2], [1, -2, 5])
Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, -1, 3], [-1, 1, -2], [1, -2, 5]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-quotient · candidate
30. Quotient Dimension with Ordered Exponent Subtraction
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[1, -1, -2], denominator=[2, 2, -2]. Give (numerator vector, denominator vector, quotient exponent vector).
View answer and explanation
([1, -1, -2], [2, 2, -2], [-1, -3, 0])
Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, -1, -2], [2, 2, -2], [-1, -3, 0]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-quotient · candidate
31. Quotient Dimension with Ordered Exponent Subtraction
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[2, -1, 0], denominator=[-2, -2, -1]. Give (numerator vector, denominator vector, quotient exponent vector).
View answer and explanation
([2, -1, 0], [-2, -2, -1], [4, 1, 1])
Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, -1, 0], [-2, -2, -1], [4, 1, 1]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-quotient · candidate
32. Quotient Dimension with Ordered Exponent Subtraction
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[3, -1, 2], denominator=[1, -1, -1]. Give (numerator vector, denominator vector, quotient exponent vector).
View answer and explanation
([3, -1, 2], [1, -1, -1], [2, 0, 3])
Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, -1, 2], [1, -1, -1], [2, 0, 3]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-quotient · candidate
33. Quotient Dimension with Ordered Exponent Subtraction
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[-3, 0, -3], denominator=[-3, 0, -1]. Give (numerator vector, denominator vector, quotient exponent vector).
View answer and explanation
([-3, 0, -3], [-3, 0, -1], [0, 0, -2])
Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 0, -3], [-3, 0, -1], [0, 0, -2]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-quotient · candidate
34. Quotient Dimension with Ordered Exponent Subtraction
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[-2, 0, -1], denominator=[0, 1, -1]. Give (numerator vector, denominator vector, quotient exponent vector).
View answer and explanation
([-2, 0, -1], [0, 1, -1], [-2, -1, 0])
Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 0, -1], [0, 1, -1], [-2, -1, 0]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-quotient · candidate
35. Quotient Dimension with Ordered Exponent Subtraction
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[-1, 0, 1], denominator=[3, 2, -1]. Give (numerator vector, denominator vector, quotient exponent vector).
View answer and explanation
([-1, 0, 1], [3, 2, -1], [-4, -2, 2])
Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 0, 1], [3, 2, -1], [-4, -2, 2]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-quotient · candidate
36. Quotient Dimension with Ordered Exponent Subtraction
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[0, 0, 3], denominator=[-1, -2, 0]. Give (numerator vector, denominator vector, quotient exponent vector).
View answer and explanation
([0, 0, 3], [-1, -2, 0], [1, 2, 3])
Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 0, 3], [-1, -2, 0], [1, 2, 3]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-quotient · candidate
37. Quotient Dimension with Ordered Exponent Subtraction
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[1, 0, -2], denominator=[2, -1, 0]. Give (numerator vector, denominator vector, quotient exponent vector).
View answer and explanation
([1, 0, -2], [2, -1, 0], [-1, 1, -2])
Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 0, -2], [2, -1, 0], [-1, 1, -2]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-quotient · candidate
38. Quotient Dimension with Ordered Exponent Subtraction
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[2, 0, 0], denominator=[-2, 0, 0]. Give (numerator vector, denominator vector, quotient exponent vector).
View answer and explanation
([2, 0, 0], [-2, 0, 0], [4, 0, 0])
Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 0, 0], [-2, 0, 0], [4, 0, 0]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-quotient · candidate
39. Quotient Dimension with Ordered Exponent Subtraction
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[3, 0, 2], denominator=[1, 1, 0]. Give (numerator vector, denominator vector, quotient exponent vector).
View answer and explanation
([3, 0, 2], [1, 1, 0], [2, -1, 2])
Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 0, 2], [1, 1, 0], [2, -1, 2]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-quotient · candidate
40. Quotient Dimension with Ordered Exponent Subtraction
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[-3, 1, -3], denominator=[-3, 2, 0]. Give (numerator vector, denominator vector, quotient exponent vector).
View answer and explanation
([-3, 1, -3], [-3, 2, 0], [0, -1, -3])
Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 1, -3], [-3, 2, 0], [0, -1, -3]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-quotient · candidate
41. Quotient Dimension with Ordered Exponent Subtraction
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[-2, 1, -1], denominator=[0, -2, 1]. Give (numerator vector, denominator vector, quotient exponent vector).
View answer and explanation
([-2, 1, -1], [0, -2, 1], [-2, 3, -2])
Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 1, -1], [0, -2, 1], [-2, 3, -2]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-quotient · candidate
42. Quotient Dimension with Ordered Exponent Subtraction
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[-1, 1, 1], denominator=[3, -1, 1]. Give (numerator vector, denominator vector, quotient exponent vector).
View answer and explanation
([-1, 1, 1], [3, -1, 1], [-4, 2, 0])
Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 1, 1], [3, -1, 1], [-4, 2, 0]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-quotient · candidate
43. Quotient Dimension with Ordered Exponent Subtraction
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[0, 1, 3], denominator=[-1, 0, 1]. Give (numerator vector, denominator vector, quotient exponent vector).
View answer and explanation
([0, 1, 3], [-1, 0, 1], [1, 1, 2])
Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 1, 3], [-1, 0, 1], [1, 1, 2]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-quotient · candidate
44. Quotient Dimension with Ordered Exponent Subtraction
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[1, 1, -2], denominator=[2, 1, 1]. Give (numerator vector, denominator vector, quotient exponent vector).
View answer and explanation
([1, 1, -2], [2, 1, 1], [-1, 0, -3])
Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 1, -2], [2, 1, 1], [-1, 0, -3]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-quotient · candidate
45. Quotient Dimension with Ordered Exponent Subtraction
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[2, 1, 0], denominator=[-2, 2, 1]. Give (numerator vector, denominator vector, quotient exponent vector).
View answer and explanation
([2, 1, 0], [-2, 2, 1], [4, -1, -1])
Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 1, 0], [-2, 2, 1], [4, -1, -1]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-quotient · candidate
46. Quotient Dimension with Ordered Exponent Subtraction
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[3, 1, 2], denominator=[1, -2, 2]. Give (numerator vector, denominator vector, quotient exponent vector).
View answer and explanation
([3, 1, 2], [1, -2, 2], [2, 3, 0])
Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 1, 2], [1, -2, 2], [2, 3, 0]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-quotient · candidate
47. Quotient Dimension with Ordered Exponent Subtraction
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[-3, 2, -3], denominator=[-3, -1, 2]. Give (numerator vector, denominator vector, quotient exponent vector).
View answer and explanation
([-3, 2, -3], [-3, -1, 2], [0, 3, -5])
Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 2, -3], [-3, -1, 2], [0, 3, -5]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-quotient · candidate
48. Quotient Dimension with Ordered Exponent Subtraction
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[-2, 2, -1], denominator=[0, 0, 2]. Give (numerator vector, denominator vector, quotient exponent vector).
View answer and explanation
([-2, 2, -1], [0, 0, 2], [-2, 2, -3])
Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 2, -1], [0, 0, 2], [-2, 2, -3]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-quotient · candidate
49. Quotient Dimension with Ordered Exponent Subtraction
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[-1, 2, 1], denominator=[3, 1, 2]. Give (numerator vector, denominator vector, quotient exponent vector).
View answer and explanation
([-1, 2, 1], [3, 1, 2], [-4, 1, -1])
Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 2, 1], [3, 1, 2], [-4, 1, -1]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-quotient · candidate
50. Quotient Dimension with Ordered Exponent Subtraction
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[0, 2, 3], denominator=[-1, 2, 2]. Give (numerator vector, denominator vector, quotient exponent vector).
View answer and explanation
([0, 2, 3], [-1, 2, 2], [1, 0, 1])
Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 2, 3], [-1, 2, 2], [1, 0, 1]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-quotient · candidate
51. Integer Power of a Dimension Vector
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[-3, -1, -3], exponent=-3. Give (base vector, exponent, exponent-scaled vector).
View answer and explanation
([-3, -1, -3], -3, [9, 3, 9])
Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, -1, -3], -3, [9, 3, 9]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-power · candidate
52. Integer Power of a Dimension Vector
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[-2, -1, -1], exponent=-2. Give (base vector, exponent, exponent-scaled vector).
View answer and explanation
([-2, -1, -1], -2, [4, 2, 2])
Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, -1, -1], -2, [4, 2, 2]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-power · candidate
53. Integer Power of a Dimension Vector
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[-1, -1, 1], exponent=-1. Give (base vector, exponent, exponent-scaled vector).
View answer and explanation
([-1, -1, 1], -1, [1, 1, -1])
Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, -1, 1], -1, [1, 1, -1]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-power · candidate
54. Integer Power of a Dimension Vector
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[0, -1, 3], exponent=0. Give (base vector, exponent, exponent-scaled vector).
View answer and explanation
([0, -1, 3], 0, [0, 0, 0])
Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, -1, 3], 0, [0, 0, 0]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-power · candidate
55. Integer Power of a Dimension Vector
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[1, -1, -2], exponent=1. Give (base vector, exponent, exponent-scaled vector).
View answer and explanation
([1, -1, -2], 1, [1, -1, -2])
Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, -1, -2], 1, [1, -1, -2]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-power · candidate
56. Integer Power of a Dimension Vector
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[2, -1, 0], exponent=2. Give (base vector, exponent, exponent-scaled vector).
View answer and explanation
([2, -1, 0], 2, [4, -2, 0])
Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, -1, 0], 2, [4, -2, 0]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-power · candidate
57. Integer Power of a Dimension Vector
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[3, -1, 2], exponent=3. Give (base vector, exponent, exponent-scaled vector).
View answer and explanation
([3, -1, 2], 3, [9, -3, 6])
Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, -1, 2], 3, [9, -3, 6]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-power · candidate
58. Integer Power of a Dimension Vector
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[-3, 0, -3], exponent=-3. Give (base vector, exponent, exponent-scaled vector).
View answer and explanation
([-3, 0, -3], -3, [9, 0, 9])
Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 0, -3], -3, [9, 0, 9]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-power · candidate
59. Integer Power of a Dimension Vector
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[-2, 0, -1], exponent=-2. Give (base vector, exponent, exponent-scaled vector).
View answer and explanation
([-2, 0, -1], -2, [4, 0, 2])
Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 0, -1], -2, [4, 0, 2]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-power · candidate
60. Integer Power of a Dimension Vector
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[-1, 0, 1], exponent=-1. Give (base vector, exponent, exponent-scaled vector).
View answer and explanation
([-1, 0, 1], -1, [1, 0, -1])
Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 0, 1], -1, [1, 0, -1]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-power · candidate
61. Integer Power of a Dimension Vector
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[0, 0, 3], exponent=0. Give (base vector, exponent, exponent-scaled vector).
View answer and explanation
([0, 0, 3], 0, [0, 0, 0])
Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 0, 3], 0, [0, 0, 0]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-power · candidate
62. Integer Power of a Dimension Vector
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[1, 0, -2], exponent=1. Give (base vector, exponent, exponent-scaled vector).
View answer and explanation
([1, 0, -2], 1, [1, 0, -2])
Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 0, -2], 1, [1, 0, -2]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-power · candidate
63. Integer Power of a Dimension Vector
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[2, 0, 0], exponent=2. Give (base vector, exponent, exponent-scaled vector).
View answer and explanation
([2, 0, 0], 2, [4, 0, 0])
Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 0, 0], 2, [4, 0, 0]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-power · candidate
64. Integer Power of a Dimension Vector
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[3, 0, 2], exponent=3. Give (base vector, exponent, exponent-scaled vector).
View answer and explanation
([3, 0, 2], 3, [9, 0, 6])
Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 0, 2], 3, [9, 0, 6]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-power · candidate
65. Integer Power of a Dimension Vector
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[-3, 1, -3], exponent=-3. Give (base vector, exponent, exponent-scaled vector).
View answer and explanation
([-3, 1, -3], -3, [9, -3, 9])
Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 1, -3], -3, [9, -3, 9]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-power · candidate
66. Integer Power of a Dimension Vector
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[-2, 1, -1], exponent=-2. Give (base vector, exponent, exponent-scaled vector).
View answer and explanation
([-2, 1, -1], -2, [4, -2, 2])
Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 1, -1], -2, [4, -2, 2]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-power · candidate
67. Integer Power of a Dimension Vector
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[-1, 1, 1], exponent=-1. Give (base vector, exponent, exponent-scaled vector).
View answer and explanation
([-1, 1, 1], -1, [1, -1, -1])
Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 1, 1], -1, [1, -1, -1]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-power · candidate
68. Integer Power of a Dimension Vector
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[0, 1, 3], exponent=0. Give (base vector, exponent, exponent-scaled vector).
View answer and explanation
([0, 1, 3], 0, [0, 0, 0])
Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 1, 3], 0, [0, 0, 0]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-power · candidate
69. Integer Power of a Dimension Vector
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[1, 1, -2], exponent=1. Give (base vector, exponent, exponent-scaled vector).
View answer and explanation
([1, 1, -2], 1, [1, 1, -2])
Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 1, -2], 1, [1, 1, -2]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-power · candidate
70. Integer Power of a Dimension Vector
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[2, 1, 0], exponent=2. Give (base vector, exponent, exponent-scaled vector).
View answer and explanation
([2, 1, 0], 2, [4, 2, 0])
Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 1, 0], 2, [4, 2, 0]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-power · candidate
71. Integer Power of a Dimension Vector
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[3, 1, 2], exponent=3. Give (base vector, exponent, exponent-scaled vector).
View answer and explanation
([3, 1, 2], 3, [9, 3, 6])
Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 1, 2], 3, [9, 3, 6]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-power · candidate
72. Integer Power of a Dimension Vector
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[-3, 2, -3], exponent=-3. Give (base vector, exponent, exponent-scaled vector).
View answer and explanation
([-3, 2, -3], -3, [9, -6, 9])
Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 2, -3], -3, [9, -6, 9]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-power · candidate
73. Integer Power of a Dimension Vector
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[-2, 2, -1], exponent=-2. Give (base vector, exponent, exponent-scaled vector).
View answer and explanation
([-2, 2, -1], -2, [4, -4, 2])
Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 2, -1], -2, [4, -4, 2]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-power · candidate
74. Integer Power of a Dimension Vector
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[-1, 2, 1], exponent=-1. Give (base vector, exponent, exponent-scaled vector).
View answer and explanation
([-1, 2, 1], -1, [1, -2, -1])
Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 2, 1], -1, [1, -2, -1]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-power · candidate
75. Integer Power of a Dimension Vector
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[0, 2, 3], exponent=0. Give (base vector, exponent, exponent-scaled vector).
View answer and explanation
([0, 2, 3], 0, [0, 0, 0])
Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 2, 3], 0, [0, 0, 0]).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-power · candidate
76. Equal Dimension Vectors Are Not a Physical Proof
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-3, -1, -3], right=[-2, -1, -3]. Give (left vector, right vector, identical?).
View answer and explanation
([-3, -1, -3], [-2, -1, -3], No)
Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, -1, -3], [-2, -1, -3], No).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-equation-consistency · candidate
77. Equal Dimension Vectors Are Not a Physical Proof
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-2, -1, -1], right=[-2, -1, -1]. Give (left vector, right vector, identical?).
View answer and explanation
([-2, -1, -1], [-2, -1, -1], Yes)
Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, -1, -1], [-2, -1, -1], Yes).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-equation-consistency · candidate
78. Equal Dimension Vectors Are Not a Physical Proof
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-1, -1, 1], right=[-1, -1, 1]. Give (left vector, right vector, identical?).
View answer and explanation
([-1, -1, 1], [-1, -1, 1], Yes)
Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, -1, 1], [-1, -1, 1], Yes).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-equation-consistency · candidate
79. Equal Dimension Vectors Are Not a Physical Proof
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[0, -1, 3], right=[1, -1, 3]. Give (left vector, right vector, identical?).
View answer and explanation
([0, -1, 3], [1, -1, 3], No)
Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, -1, 3], [1, -1, 3], No).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-equation-consistency · candidate
80. Equal Dimension Vectors Are Not a Physical Proof
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[1, -1, -2], right=[1, -1, -2]. Give (left vector, right vector, identical?).
View answer and explanation
([1, -1, -2], [1, -1, -2], Yes)
Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, -1, -2], [1, -1, -2], Yes).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-equation-consistency · candidate
81. Equal Dimension Vectors Are Not a Physical Proof
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[2, -1, 0], right=[2, -1, 0]. Give (left vector, right vector, identical?).
View answer and explanation
([2, -1, 0], [2, -1, 0], Yes)
Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, -1, 0], [2, -1, 0], Yes).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-equation-consistency · candidate
82. Equal Dimension Vectors Are Not a Physical Proof
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[3, -1, 2], right=[4, -1, 2]. Give (left vector, right vector, identical?).
View answer and explanation
([3, -1, 2], [4, -1, 2], No)
Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, -1, 2], [4, -1, 2], No).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-equation-consistency · candidate
83. Equal Dimension Vectors Are Not a Physical Proof
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-3, 0, -3], right=[-3, 0, -3]. Give (left vector, right vector, identical?).
View answer and explanation
([-3, 0, -3], [-3, 0, -3], Yes)
Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 0, -3], [-3, 0, -3], Yes).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-equation-consistency · candidate
84. Equal Dimension Vectors Are Not a Physical Proof
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-2, 0, -1], right=[-2, 0, -1]. Give (left vector, right vector, identical?).
View answer and explanation
([-2, 0, -1], [-2, 0, -1], Yes)
Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 0, -1], [-2, 0, -1], Yes).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-equation-consistency · candidate
85. Equal Dimension Vectors Are Not a Physical Proof
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-1, 0, 1], right=[0, 0, 1]. Give (left vector, right vector, identical?).
View answer and explanation
([-1, 0, 1], [0, 0, 1], No)
Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 0, 1], [0, 0, 1], No).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-equation-consistency · candidate
86. Equal Dimension Vectors Are Not a Physical Proof
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[0, 0, 3], right=[0, 0, 3]. Give (left vector, right vector, identical?).
View answer and explanation
([0, 0, 3], [0, 0, 3], Yes)
Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 0, 3], [0, 0, 3], Yes).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-equation-consistency · candidate
87. Equal Dimension Vectors Are Not a Physical Proof
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[1, 0, -2], right=[1, 0, -2]. Give (left vector, right vector, identical?).
View answer and explanation
([1, 0, -2], [1, 0, -2], Yes)
Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 0, -2], [1, 0, -2], Yes).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-equation-consistency · candidate
88. Equal Dimension Vectors Are Not a Physical Proof
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[2, 0, 0], right=[3, 0, 0]. Give (left vector, right vector, identical?).
View answer and explanation
([2, 0, 0], [3, 0, 0], No)
Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 0, 0], [3, 0, 0], No).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-equation-consistency · candidate
89. Equal Dimension Vectors Are Not a Physical Proof
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[3, 0, 2], right=[3, 0, 2]. Give (left vector, right vector, identical?).
View answer and explanation
([3, 0, 2], [3, 0, 2], Yes)
Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 0, 2], [3, 0, 2], Yes).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-equation-consistency · candidate
90. Equal Dimension Vectors Are Not a Physical Proof
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-3, 1, -3], right=[-3, 1, -3]. Give (left vector, right vector, identical?).
View answer and explanation
([-3, 1, -3], [-3, 1, -3], Yes)
Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 1, -3], [-3, 1, -3], Yes).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-equation-consistency · candidate
91. Equal Dimension Vectors Are Not a Physical Proof
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-2, 1, -1], right=[-1, 1, -1]. Give (left vector, right vector, identical?).
View answer and explanation
([-2, 1, -1], [-1, 1, -1], No)
Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 1, -1], [-1, 1, -1], No).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-equation-consistency · candidate
92. Equal Dimension Vectors Are Not a Physical Proof
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-1, 1, 1], right=[-1, 1, 1]. Give (left vector, right vector, identical?).
View answer and explanation
([-1, 1, 1], [-1, 1, 1], Yes)
Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 1, 1], [-1, 1, 1], Yes).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-equation-consistency · candidate
93. Equal Dimension Vectors Are Not a Physical Proof
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[0, 1, 3], right=[0, 1, 3]. Give (left vector, right vector, identical?).
View answer and explanation
([0, 1, 3], [0, 1, 3], Yes)
Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 1, 3], [0, 1, 3], Yes).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-equation-consistency · candidate
94. Equal Dimension Vectors Are Not a Physical Proof
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[1, 1, -2], right=[2, 1, -2]. Give (left vector, right vector, identical?).
View answer and explanation
([1, 1, -2], [2, 1, -2], No)
Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 1, -2], [2, 1, -2], No).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-equation-consistency · candidate
95. Equal Dimension Vectors Are Not a Physical Proof
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[2, 1, 0], right=[2, 1, 0]. Give (left vector, right vector, identical?).
View answer and explanation
([2, 1, 0], [2, 1, 0], Yes)
Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 1, 0], [2, 1, 0], Yes).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-equation-consistency · candidate
96. Equal Dimension Vectors Are Not a Physical Proof
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[3, 1, 2], right=[3, 1, 2]. Give (left vector, right vector, identical?).
View answer and explanation
([3, 1, 2], [3, 1, 2], Yes)
Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 1, 2], [3, 1, 2], Yes).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-equation-consistency · candidate
97. Equal Dimension Vectors Are Not a Physical Proof
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-3, 2, -3], right=[-2, 2, -3]. Give (left vector, right vector, identical?).
View answer and explanation
([-3, 2, -3], [-2, 2, -3], No)
Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 2, -3], [-2, 2, -3], No).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-equation-consistency · candidate
98. Equal Dimension Vectors Are Not a Physical Proof
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-2, 2, -1], right=[-2, 2, -1]. Give (left vector, right vector, identical?).
View answer and explanation
([-2, 2, -1], [-2, 2, -1], Yes)
Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 2, -1], [-2, 2, -1], Yes).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-equation-consistency · candidate
99. Equal Dimension Vectors Are Not a Physical Proof
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-1, 2, 1], right=[-1, 2, 1]. Give (left vector, right vector, identical?).
View answer and explanation
([-1, 2, 1], [-1, 2, 1], Yes)
Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 2, 1], [-1, 2, 1], Yes).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-equation-consistency · candidate
100. Equal Dimension Vectors Are Not a Physical Proof
Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[0, 2, 3], right=[1, 2, 3]. Give (left vector, right vector, identical?).
View answer and explanation
([0, 2, 3], [1, 2, 3], No)
Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 2, 3], [1, 2, 3], No).
Reviewed step-by-step derivation
Dimensional analysis · qf-app20-dimension-equation-consistency · candidate
Canonical question JSONL · English JSONL · Validation report