// Numbas version: finer_feedback_settings
{"name": "Vectors: Vector Product 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Vectors: Vector Product 2", "tags": [], "metadata": {"description": "<p>Find a perpendicular vector to a pair of vectors.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Find a vector perpendicular to both $\\mathbf p$ and $\\mathbf q$, where \\[\\mathbf p = \\var{p},\\,\\mathbf q = \\var{q}.\\]</p>", "advice": "<p>If we want to find a vector perpendicular to both $\\mathbf p$ and $\\mathbf q$, this is done by the taking the vector product between $\\mathbf p$ and $\\mathbf q$.</p>\n<p>(Note: This can be either $\\simplify{cross(mathbf:p,mathbf:q)}$ or&nbsp;$\\simplify{cross(mathbf:q,mathbf:p)}$, both will produce a vector perpendicular to $\\mathbf p$ and $\\mathbf q$, they will just be in opposite directions.)</p>\n<p>Taking the vector product between $\\mathbf p$ and $\\mathbf q$:</p>\n<p></p>\n<p>\\[ \\begin{split}\\simplify{cross(mathbf:p,mathbf:q)} &amp;\\,= \\begin{vmatrix} \\mathbf i &amp;\\mathbf j &amp;\\mathbf k \\\\ \\var{p[0]} &amp;\\var{p[1]} &amp;\\var{p[2]} \\\\ \\var{q[0]} &amp;\\var{q[1]} &amp;\\var{q[2]} \\end{vmatrix}\\\\ \\\\&amp;\\,= (\\simplify[!collectNumbers]{{p[1]}*{q[2]}-{p[2]}*{q[1]}}) \\mathbf i -&nbsp;(\\simplify[!collectNumbers]{{p[0]}*{q[2]}-{p[2]}*{q[0]}}) \\mathbf j +&nbsp;(\\simplify[!collectNumbers]{{p[0]}*{q[1]}-{p[1]}*{q[0]}}) \\mathbf k \\\\\\\\ &amp;\\,=&nbsp;\\simplify[all,!noLeadingMinus]{{c1[0]}*mathbf:i+ {c1[1]}*mathbf:j + {c1[2]}*mathbf:k}\\end{split} \\]</p>\n<p>Therefore,</p>\n<p>\\[ \\simplify{cross(mathbf:p,mathbf:q) = {c1}} .\\]</p>\n<p></p>\n<p>(NOTE: The vector calculated here is&nbsp;<strong></strong><em>one&nbsp;</em><em>solution,&nbsp;</em>but any scalar multiple of $\\simplify{cross(mathbf:p,mathbf:q)}$ will also be perpendicular to both $\\mathbf p$ and $\\mathbf q$.)</p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"p": {"name": "p", "group": "Ungrouped variables", "definition": "vector(random(-2..6),random(-4..4 except 0),random(1..7))", "description": "", "templateType": "anything", "can_override": false}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "vector(random(-2..5),random(-4..4 except [-1,0,1]),random(1..5))", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "cross(p,q)", "description": "", "templateType": "anything", "can_override": false}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "cross(q,p)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["p", "q", "c1", "c2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>[[0]]</p>", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "correctAnswer": "vector(c2[0],c2[1],c2[2])", "correctAnswerFractions": false, "numRows": "3", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "correctAnswer": "vector(c1[0],c1[1],c1[2])", "correctAnswerFractions": false, "numRows": "3", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}], "resources": []}]}], "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}