# Question 35959: Vector Equality in a Fixed Basis

[Return to catalogue page 180](https://ocme.evemisslab.com/en/questions/page/180/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261003-advance-20261004-r05-vector-foundations/#q-20261003-advance-20261004-r05-vector-0084)

This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0070-35959 · q-20261003-advance-20261004-r05-vector-0084



## Original question and givens

In the fixed basis of the real plane R², u=(-4, 1) and v=(-4, 1). Are u and v equal?


- G001 — In the fixed basis of the real plane R², u=(-4, 1) and v=(-4, 1). Are u and v equal?





## 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.


- [mko-vector](https://ocme.evemisslab.com/en/mko/mko-vector/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [qf-vector-coordinate-equality, [domain, operation, u, v]]

After

derived = [vector, vector_equality, reals]

Conditions: none

Use THIS bank’s actual fields,operation/domain and types.BASE retains permissible extra metadata.Do not import LDL SAFE/set limits or a different bank’s deduplication rules.

Step identifiers and dependenciesRule: R70_SOURCE_bank_specific_interface · Uses: G001





S002 · impliesBefore

source roles = [[-4, 1], [-4, 1], null]

After

derived = [two real coordinates in ONE ordered basis]

Conditions: none

The vectors use ONE ordered basis.Type ALL TWO RAW SAFE components and k.These are vector coordinates,not points,lengths or a dot product.Component sums alone do not determine equality.

Step identifiers and dependenciesRule: Vector_GIVEN_common_ordered_basis_RAW_two_SAFE_components · Uses: G001, S001





S003 · impliesBefore

source roles = [0, -4, -4]

After

derived = [Yes]

Conditions: none

Compare EACH corresponding position without permuting coordinates.Vector equality requires BOTH components to match.

Step identifiers and dependenciesRule: Vector_EQUALITY_EACH_corresponding_coordinate · Uses: G001, S002





S004 · impliesBefore

source roles = [1, 1, 1]

After

derived = [Yes]

Conditions: none

Compare EACH corresponding position without permuting coordinates.Vector equality requires BOTH components to match.

Step identifiers and dependenciesRule: Vector_EQUALITY_EACH_corresponding_coordinate · Uses: G001, S003





S005 · impliesBefore

source roles = [[Yes, Yes]]

After

derived = [Yes]

Conditions: none

Conjoin BOTH coordinate equalities;neither one match nor equal component sums is enough.

Step identifiers and dependenciesRule: Vector_EQUALITY_conjunction_not_component_sum · Uses: G001, S004









## Reverse checks


- S005 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
- S004 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
- S003 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
- S002 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
- S001 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.





## Result

Yes

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: 759931bb86b618536eb90fb874bdc61b566f407a9359c6e45fb198acb4e38afd
- candidate record SHA-256: a30b71a8f1009266e648b5fc14395a748e67bd1ed121af10ec7ea9966841ecf1
- semantic review receipt SHA-256: 9820ac6529139bd53d516c8bfdfdd41184b0cd52dad69fa433344010db37af80
- question-first witness SHA-256: c9c939762bbd4d3ea3403084c8e95541dc2c7e56eb82fc74eaba8c80a4517dab
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261003-advance-20261004-r05-vector-0084.json)