{"query": "Vectors and matrices: a dot product and a determinant", "count": 20, "results": [{"id": "card_works_linear_algebra", "title": "Vectors and matrices: a dot product and a determinant", "shelf": "the-works", "surface": "secular", "snippet": "The dot product of two 3-vectors measures how they align; the determinant of a 2×2 matrix measures the signed area it scales. Both computed exactly.  Worked & sealed by the engine — [1,2,3]·[4,5,6] = ", "authority_tier": "verified", "source": "The Works — worked & sealed", "readable": false, "generated": false}, {"id": "card_calc_v_matrix_determinant", "title": "Determinant", "shelf": "calculations", "surface": "secular", "snippet": "Determinant — linear_algebra. Formula: det A. Canonical FORM: linear_system (A x = b) — engine verifier matrix.determinant — the deterministic check the concordance runs. Same form, different domain: ", "authority_tier": "reference", "source": "The Calculation Map — every calculation, mapped by form", "readable": false, "generated": false}, {"id": "card_src_openstax_calculus_volume_3_2_4_the_cross_product_d53b2426", "title": "2.4 The Cross Product — Calculus Volume 3", "shelf": "mathematics", "surface": "secular", "snippet": "2.4\n\nThe Cross Product\n\nLearning Objectives\n\n2.4.1\nCalculate the cross product of two given vectors.\n\n2.4.2\nUse determinants to calculate a cross product.\n\n2.4.3\nFind a vector orthogonal to two given ", "authority_tier": "reference", "source": "OpenStax: Calculus Volume 3 (CC-BY 4.0)", "readable": false, "generated": false}, {"id": "card_src_openstax_calculus_volume_3_key_terms_508c1c4e", "title": "Key Terms — Calculus Volume 3", "shelf": "mathematics", "surface": "secular", "snippet": "Key Terms\n\ncomponenta scalar that describes either the vertical or horizontal direction of a vector\n\ncoordinate plane\na plane containing two of the three coordinate axes in the three-dimensional coord", "authority_tier": "reference", "source": "OpenStax: Calculus Volume 3 (CC-BY 4.0)", "readable": false, "generated": false}, {"id": "card_v_06eca966d113", "title": "[1,2,3]·[4,5,6] = 4+10+18 = 32 · det [[1,2],[3,4]] = 1·4 − 2·3 = −2", "shelf": "linear_algebra", "surface": "secular", "snippet": "Verdict: HOLDS.\n• vector.dot_product: [1.0, 2.0, 3.0] · [4.0, 5.0, 6.0] = 32.0\n• matrix.determinant: det = -2\nSealed and independently re-checkable: https://narrowhighway.com/s/5a497c248aa2ecb6cdbbda7", "authority_tier": "engine", "source": "Verified by the engine — linear_algebra", "readable": false, "generated": false}, {"id": "card_src_etym_determinant", "title": "determinant", "shelf": "etymology", "surface": "secular", "snippet": "determinant: etymology (Webster 1913) — a.: [L. determinans, p. pr. of determinare: cf. F. déterminant.]. From Webster's Revised Unabridged Dictionary (1913), public domain.", "authority_tier": "reference", "source": "Webster's Revised Unabridged Dictionary (1913), Project Gutenberg eBook #29765 — public domain", "readable": true, "generated": false}, {"id": "card_src_pron_determinant", "title": "determinant", "shelf": "pronunciation", "surface": "secular", "snippet": "determinant: pronounced (ARPABET) D IH0 T ER1 M AH0 N AH0 N T. From the CMU Pronouncing Dictionary — the standard machine-readable pronunciations of North American English.", "authority_tier": "reference", "source": "CMU Pronouncing Dictionary (cmudict) — BSD-2-Clause, Carnegie Mellon", "readable": false, "generated": false}, {"id": "card_src_pron_determinants", "title": "determinants", "shelf": "pronunciation", "surface": "secular", "snippet": "determinants: pronounced (ARPABET) D IH0 T ER1 M AH0 N AH0 N T S. From the CMU Pronouncing Dictionary — the standard machine-readable pronunciations of North American English.", "authority_tier": "reference", "source": "CMU Pronouncing Dictionary (cmudict) — BSD-2-Clause, Carnegie Mellon", "readable": false, "generated": false}, {"id": "card_seal_e4ff8c7f3e6418c46ffa1e85d9371baff23a4c7991eb1f89ae5c92d1d89c71ba", "title": "Receipt e4ff8c7f3e64… — sealed: record", "shelf": "seals", "surface": "secular", "snippet": "{\n \"anchors\": [],\n \"axis_coords\": {\n  \"axis\": \"mathematics\",\n  \"dimensions\": [\n   \"reasoning\"\n  ]\n },\n \"gate_results\": [\n  {\n   \"details\": {\n    \"broken_at\": null,\n    \"confirmed_steps\": 1,\n    \"error", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "readable": false, "generated": false}, {"id": "card_seal_ed5b9f0c5b9810c430efbf21b7eb3f699dad98958767401afbe6f6356b3d885f", "title": "Receipt ed5b9f0c5b98… — sealed: record", "shelf": "seals", "surface": "secular", "snippet": "{\n \"anchors\": [],\n \"axis_coords\": {\n  \"axis\": \"mathematics\",\n  \"dimensions\": [\n   \"reasoning\"\n  ]\n },\n \"gate_results\": [\n  {\n   \"details\": {\n    \"broken_at\": null,\n    \"confirmed_steps\": 1,\n    \"error", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "readable": false, "generated": false}, {"id": "card_seal_adf72520318821c22e8725ce4c3b781c2947ad2ef8b2d3112187b7823ed3eacc", "title": "Receipt adf725203188… — sealed: record", "shelf": "seals", "surface": "secular", "snippet": "{\n \"anchors\": [],\n \"axis_coords\": {\n  \"axis\": \"linear_algebra\",\n  \"dimensions\": [\n   \"physical_substance\",\n   \"reasoning\"\n  ]\n },\n \"gate_results\": [\n  {\n   \"details\": {\n    \"broken_at\": null,\n    \"con", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "readable": false, "generated": false}, {"id": "card_seal_abf996efaf2f05a126f5dd6dc2009e0004d5562047585459313907f1eb1fdca7", "title": "Receipt abf996efaf2f… — sealed: record", "shelf": "seals", "surface": "secular", "snippet": "{\n \"anchors\": [],\n \"axis_coords\": {\n  \"axis\": \"linear_algebra\",\n  \"dimensions\": [\n   \"physical_substance\",\n   \"reasoning\"\n  ]\n },\n \"gate_results\": [\n  {\n   \"details\": {\n    \"broken_at\": null,\n    \"con", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "readable": false, "generated": false}, {"id": "card_seal_d9c9da87797c35ab8271fefa6d09a5886ee65816f685b734ea565fc1deae601e", "title": "Receipt d9c9da87797c… — sealed: record", "shelf": "seals", "surface": "secular", "snippet": "{\n \"anchors\": [],\n \"axis_coords\": {\n  \"axis\": \"mathematics\",\n  \"dimensions\": [\n   \"reasoning\"\n  ]\n },\n \"gate_results\": [\n  {\n   \"details\": {\n    \"broken_at\": null,\n    \"confirmed_steps\": 1,\n    \"error", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "readable": false, "generated": false}, {"id": "card_seal_65260aca3da7350a47bfef8f9df8a500976d9e11b239a56b97fd2cfe3214ea8e", "title": "Receipt 65260aca3da7… — sealed: record", "shelf": "seals", "surface": "secular", "snippet": "{\n \"anchors\": [],\n \"axis_coords\": {\n  \"axis\": \"mathematics\",\n  \"dimensions\": [\n   \"reasoning\"\n  ]\n },\n \"gate_results\": [\n  {\n   \"details\": {\n    \"broken_at\": null,\n    \"confirmed_steps\": 1,\n    \"error", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "readable": false, "generated": false}, {"id": "card_seal_a8b8b0dd40516a1b1432c112a3ddce1781ba924fc8fcb975a0f2d943bf702c03", "title": "Receipt a8b8b0dd4051… — sealed: record", "shelf": "seals", "surface": "secular", "snippet": "{\n \"anchors\": [],\n \"axis_coords\": {\n  \"axis\": \"linear_algebra\",\n  \"dimensions\": [\n   \"physical_substance\",\n   \"reasoning\"\n  ]\n },\n \"gate_results\": [\n  {\n   \"details\": {\n    \"broken_at\": null,\n    \"con", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "readable": false, "generated": false}, {"id": "card_src_openstax_university_physics_volume_1_2_4_products_of_vectors_062a33ff", "title": "2.4 Products of Vectors — University Physics Volume 1", "shelf": "physics", "surface": "secular", "snippet": "2.4\n\nProducts of Vectors\n\nLearning Objectives\nBy the end of this section, you will be able to:\n\nExplain the difference between the scalar product and the vector product of two vectors.\n\nDetermine the ", "authority_tier": "reference", "source": "OpenStax: University Physics Volume 1 (CC-BY 4.0)", "readable": false, "generated": false}, {"id": "card_theory_linear_algebra", "title": "Linear algebra (vector spaces, eigenvalues)", "shelf": "theories", "surface": "secular", "snippet": "Linear algebra (vector spaces, eigenvalues) — an engine domain that can touch it: linear_algebra. 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