// Numbas version: finer_feedback_settings
{"name": "Vectors: Vector Product 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Vectors: Vector Product 1", "tags": [], "metadata": {"description": "<p>Calculate the vector product between two vectors.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Given the vectors $\\mathbf a = \\var{a}$ and $\\mathbf b = \\var{b}$, calculate $\\mathbf a \\, \\times \\mathbf b$.</p>", "advice": "<p>To find the vector product of the vectors $\\mathbf a$ and $\\mathbf b$, we can use the formula</p>\n<p>\\[ \\simplify{cross(mathbf:a,mathbf:b)}&nbsp;= (a_2 b_3 - a_3 b_2)\\mathbf i - (a_1 b_3 - a_3 b_1) \\mathbf j + (a_1 b_2 - a_2 b_1)\\mathbf k.\\]</p>\n<p>NOTE: This is equivalent to finding the determinant of the matrix \\[ \\begin{pmatrix} \\mathbf i &amp;\\mathbf j &amp;\\mathbf k \\\\ a_1&nbsp; &amp;a_2 &amp;a_3 \\\\ b_1 &amp;b_2 &amp;b_3 \\end{pmatrix}\\]</p>\n<p>So, for the vectors&nbsp;$\\mathbf a = \\var{a}$ and $\\mathbf b = \\var{b}$,</p>\n<p>\\[ \\begin{split}\\simplify{cross(mathbf:a,mathbf:b)} &amp;\\,= \\begin{vmatrix} \\mathbf i &amp;\\mathbf j &amp;\\mathbf k \\\\ \\var{a[0]} &amp;\\var{a[1]} &amp;\\var{a[2]} \\\\ \\var{b[0]} &amp;\\var{b[1]} &amp;\\var{b[2]} \\end{vmatrix}\\\\ \\\\&amp;\\,= (\\simplify[!collectNumbers]{{a[1]}*{b[2]}-{a[2]}*{b[1]}}) \\mathbf i -&nbsp;(\\simplify[!collectNumbers]{{a[0]}*{b[2]}-{a[2]}*{b[0]}}) \\mathbf j +&nbsp;(\\simplify[!collectNumbers]{{a[0]}*{b[1]}-{a[1]}*{b[0]}}) \\mathbf k \\\\\\\\ &amp;\\,=&nbsp;\\simplify[all,!noLeadingMinus]{{c[0]}*mathbf:i+ {c[1]}*mathbf:j + {c[2]}*mathbf:k}\\end{split} \\]</p>\n<p>Therefore,</p>\n<p>\\[ \\simplify{cross(mathbf:a,mathbf:b) = {c}} .\\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "vector(random(-2..6),random(-4..4 except 0),random(1..7))", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "vector(random(-2..5),random(-4..4 except 0),random(1..4))", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "cross(a,b)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c"], "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>$\\mathbf a \\, \\times \\mathbf b =$ [[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, "correctAnswer": "vector(c[0],c[1],c[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/"}]}]}], "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}