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Question 7979: Ordered Weighted Vector Combination

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0014-07979 · q-20261012-advance-20261016-r01-vector-0004

Original question and givens

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[2, -3], v=[1, 4], a=9007199254740991, b=-9007199254740991. Give (a*u, b*v, a*u+b*v) in that order.

  • G001 — Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[2, -3], v=[1, 4], a=9007199254740991, b=-9007199254740991. Give (a*u, b*v, a*u+b*v) in that order.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

fixed standard basis;two ordered integer coordinates

After

componentwise operations;not length or inner product

Conditions: none

Work coordinate by coordinate in the same fixed basis.Swapping coordinates,taking a norm or computing an inner product is not the requested vector answer.

Step identifiers and dependencies

Rule: ordered_vector_scope · Uses: G001

S002 · calculation

Before

au[0] := mul(9007199254740991, 2)

After

au[0] = 18014398509481982

Conditions: none

Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

au[1] := mul(9007199254740991, -3)

After

au[1] = -27021597764222973

Conditions: none

Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

bv[0] := mul(-9007199254740991, 1)

After

bv[0] = -9007199254740991

Conditions: none

Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

bv[1] := mul(-9007199254740991, 4)

After

bv[1] = -36028797018963964

Conditions: none

Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

au+bv[0] := sum(18014398509481982, -9007199254740991)

After

au+bv[0] = 9007199254740991

Conditions: none

Add the already aligned component or position,without mixing coordinates.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

au+bv[1] := sum(-27021597764222973, -36028797018963964)

After

au+bv[1] = -63050394783186937

Conditions: none

Add the already aligned component or position,without mixing coordinates.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

au=[18014398509481982, -27021597764222973];bv=[-9007199254740991, -36028797018963964]

After

value=[9007199254740991, -63050394783186937]

Conditions: none

Each weight multiplies its own vector before aligned coordinates are added.Negative weights and the zero vector use the same definition.

Step identifiers and dependencies

Rule: weighted_vector_components · Uses: G001, S007

Reverse checks

  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Recompute au+bv[1] with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute au+bv[0] with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute bv[1] with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute bv[0] with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute au[1] with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute au[0] with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([18014398509481982, -27021597764222973], [-9007199254740991, -36028797018963964], [9007199254740991, -63050394783186937])

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 88cba0451bca2db9779f4332225b9ece62cc8688d0502e8f2e78c259faab795f
  • candidate record SHA-256: 028cf3b4f5ecf660f7db8f358183ae5ddc6b6d120aed103861b02007e0f0ab7c
  • semantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77
  • question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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