{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"math","formal_name":"MATH","introduction":"MATH organises 12,500 high-school competition problems into seven subjects and five difficulty levels, each with a full worked solution. It underlies much of the later mathematics evaluation work; this catalogue imports the full test split: 5,000 problems across all seven subjects.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://github.com/hendrycks/math","indexing_mode":"index","profile":{"resources":[],"task_format":"","scoring":"","metric":"","size":"","answer_access":"","license":"","citation":"","maintainer":"","released":"","why_hard":"","related":[],"history":"","construction":"","limitations":"","saturation":"","scores":[]}},"task_id":"bd08e695-d67b-578a-98c0-aeca2328ffbc","task_key":"precalculus--test--bd08e695-d67b-578a-98c0-aeca2328ffbc","task_revision_id":"1","upstream_id":"","short_description":"The matrix for reflecting over a certain line $\\ell,$ which passes through the…","config":"precalculus","split":"test","body":"{\"level\":\"Level 5\",\"problem\":\"The matrix for reflecting over a certain line $\\\\ell,$ which passes through the origin, is given by\\n\\\\[\\\\renewcommand{\\\\arraystretch}{1.5} \\\\begin{pmatrix} -\\\\frac{2}{3} & -\\\\frac{2}{3} & -\\\\frac{1}{3} \\\\\\\\ -\\\\frac{2}{3} & \\\\frac{1}{3} & \\\\frac{2}{3} \\\\\\\\ -\\\\frac{1}{3} & \\\\frac{2}{3} & -\\\\frac{2}{3} \\\\end{pmatrix} \\\\renewcommand{\\\\arraystretch}{1}.\\\\]Find the direction vector of line $\\\\ell.$  Enter your answer in the form $\\\\begin{pmatrix} a \\\\\\\\ b \\\\\\\\ c \\\\end{pmatrix},$ where $a,$ $b,$ and $c$ are integers, $a > 0,$ and $\\\\gcd(|a|,|b|,|c|) = 1.$\",\"type\":\"Precalculus\"}","display_format":"math","language":"","answer_status":"published","assets":[],"source_url":"https://github.com/hendrycks/math","history":"initial import","indexing_mode":"index","subproblems":[],"grids":[]}