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{"name": "Parallel vectors", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Parallel vectors", "tags": [], "metadata": {"description": "<p>Determine for which value of \\(t\\) two vectors are parallel. In the first part, there is no real number \\(t\\) to make it work. In the second part, a value can be worked out.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>This question is about parallel vectors in \\(\\mathbb{R}^2\\).</p>", "advice": "<p>Two vectors \\(u\\) and \\(v\\) are parallel if there is some scalar \\(\\lambda\\) such that \\(v=\\lambda u\\) (or the other way around).</p>\n<p><strong>a)</strong> To get \\(\\begin{pmatrix}\\var{a} \\\\ t^2\\end{pmatrix}\\) parallel to the vector \\(\\var{u}\\), we would need \\(\\lambda=\\var{lambda}\\) to make it work for the first entry. However, there is no (real number) \\(t\\) such that \\(t^2=-\\var{lambda}\\). So the answer is NA.</p>\n<p>b)<strong> </strong>To get \\(\\begin{pmatrix} \\var{lambda}t \\\\ \\var{b} \\end{pmatrix}\\) parallel to the vector \\(\\var{u}\\), we need \\(\\lambda = \\var{-b}\\) to make it work for the second entry:</p>\n<p>\\[\\begin{pmatrix}\\var{lambda}t\\\\\\var{b}\\end{pmatrix}=\\var{-b}\\var{u}\\]</p>\n<p>Then \\(t= \\frac{\\var{-b*lambda*a}}{\\var{lambda}}=\\var{-b*a}\\).</p>", "rulesets": {}, "extensions": [], "variables": {"u": {"name": "u", "group": "Ungrouped variables", "definition": "vector(lambda*a,-1)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "", "templateType": "anything"}, "lambda": {"name": "lambda", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "a<>b", "maxRuns": "100"}, "ungrouped_variables": ["u", "a", "b", "lambda"], "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>For which value of \\(t\\), if any, is the vector \\(\\begin{pmatrix}\\var{a} \\\\ t^2\\end{pmatrix}\\) parallel to the vector \\(\\var{u}\\)?</p>\n<p>Enter NA if no such value exists. \\(t= \\) [[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "NA", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "na", "value": ""}]}], "sortAnswers": false}, {"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>For which value of \\(t\\), if any, is the vector \\(\\begin{pmatrix} \\var{lambda}t \\\\ \\var{b} \\end{pmatrix}\\) parallel to the vector \\(\\var{u}\\)?</p>\n<p>Enter NA if no such value exists. \\(t= \\) [[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{-b*a}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.001", "failureRate": "1", "vsetRangePoints": "5", "vsetRange": ["0", "1"], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "maxlength": {"length": "0", "partialCredit": "0", "message": ""}, "minlength": {"length": "0", "partialCredit": "0", "message": ""}, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Julia Goedecke", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5121/"}]}]}], "contributors": [{"name": "Julia Goedecke", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5121/"}]}