// Numbas version: finer_feedback_settings
{"name": "LV10 Cross or vector product of two vectors", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "LV10 Cross or vector product of two vectors", "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>The vectors $\\bf a$ and $\\bf b$ are defined as follows,<br/>$$<br/>{\\bf a} = \\var{a} \\qquad \\text{and} \\qquad \\bf b = \\var{b}.<br/>$$</p>", "advice": "<p>The vector product (also referred to as the cross product) of two vectors can be calculated as follows:</p>\n<p>$$<br/>\\pmatrix{a_1 \\\\ a_2 \\\\ a_3} \\times \\pmatrix{b_1 \\\\ b_2 \\\\ b_3} = \\pmatrix{a_2b_3-a_3b_2 \\\\ -(a_1b_3-a_3b_1) \\\\ a_1b_2-a_2b_1}.<br/>$$</p>\n<p>A way to think of this is as the following determinant:</p>\n<p>$$<br/>\\begin{vmatrix}<br/>\\bf i &amp; \\bf j &amp; \\bf k\\\\<br/>a_1 &amp; a_2 &amp; a_3 \\\\<br/>b_1 &amp; b_2 &amp; b_3<br/>\\end{vmatrix},<br/>$$</p>\n<p>where $\\bf i$, $\\bf j $, and $\\bf k$ are the standard unit basis vectors.</p>\n<p>In this question we therefore have:</p>\n<p>$$<br/>\\begin{align*}<br/>\\var{a} \\times \\var{b} &amp; = \\pmatrix{\\var{a[1]} \\times \\var{b[2]} - \\var{a[2]} \\times \\var{b[1]} \\\\ -(\\var{a[0]} \\times \\var{b[2]} - \\var{a[2]} \\times \\var{b[0]}) \\\\ \\var{a[0]} \\times \\var{b[1]} - \\var{a[1]} \\times \\var{b[0]}}\\\\<br/>&amp; = \\var{answer}.<br/>\\end{align*}<br/>$$</p>\n<p><strong><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/next-steps#Vectors\" target=\"_blank\" rel=\"noopener\">Use this link to find some resources which will help you revise this topic.</a></strong></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "vector(repeat(random(-9..9),3))", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "vector(repeat(random(-9..9),3))", "description": "", "templateType": "anything", "can_override": false}, "answer": {"name": "answer", "group": "Ungrouped variables", "definition": "cross(a,b)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "answer"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"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, "prompt": "<p>Find $\\bf a \\times \\bf b$.</p>", "correctAnswer": "answer", "correctAnswerFractions": false, "numRows": "3", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "resources": []}]}], "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}]}