// Numbas version: finer_feedback_settings
{"name": "Vectors: Parallel Vectors 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Vectors: Parallel Vectors 1", "tags": [], "metadata": {"description": "<p class=\"p1\">Finding a vector when given the magnitude of the vector and a parallel vector.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>A vector $\\mathbf a$ is parellel to $\\mathbf b = \\var{b}$, and the magnitude of $\\mathbf a$ is $|\\mathbf a| = \\var{len(a)}$. Find the vector $\\mathbf a$.</p>", "advice": "<p>In order to find $\\mathbf a$, we want to make use of $\\mathbf a$ being parellel to the vector $\\mathbf b$, and $|\\mathbf a| = \\var{len(a)}$.</p>\n<p>Since $\\mathbf a$ and $\\mathbf b$ are parellel, we can express $\\mathbf a$ in terms of $\\mathbf b$:</p>\n<p>\\[ \\mathbf a = \\lambda\\var{b},\\]</p>\n<p>where $\\lambda$ is a constant ($\\lambda \\neq 0$).</p>\n<p>Let's now express the magnitude of $\\mathbf a$ in terms of the magnitude of $\\mathbf b$:</p>\n<p>\\[ \\begin{split} |\\mathbf a| &amp;\\,= \\lambda |\\mathbf b| \\\\ &amp;\\,=\\lambda\\sqrt{(\\var{b[0]})^2 +(\\var{b[1]})^2} \\\\ &amp;\\,= \\lambda\\sqrt{\\simplify{{b[0]^2+b[1]^2}}} \\\\ &amp;\\,= \\var{len(b)}\\lambda \\end{split} \\]</p>\n<p>Since we know $|\\mathbf a| = \\var{len(a)}$ we can equate these results to find $\\lambda$:</p>\n<p>\\[ \\begin{split} \\var{len(b)} \\lambda &amp;\\,= \\var{len(a)} \\\\ \\lambda &amp;\\,= \\simplify[fractionNumbers]{{len(a)/len(b)}}. \\end{split} \\]</p>\n<p>Therefore,</p>\n<p>\\[ \\mathbf a = \\simplify{1/{c}} \\var{b} = \\var{a}. \\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(vector(sgn1*3,sgn2*4),vector(sgn1*4,sgn2*3),vector(sgn1*12,sgn2*5),vector(sgn1*5,sgn2*12),vector(sgn1*8,sgn2*15),vector(sgn1*15,sgn2*8))", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "c*a", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "sgn1": {"name": "sgn1", "group": "Ungrouped variables", "definition": "random(-1,1)", "description": "", "templateType": "anything", "can_override": false}, "sgn2": {"name": "sgn2", "group": "Ungrouped variables", "definition": "random(-1,1)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "sgn1", "sgn2"], "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 = $[[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(a[0],a[1])", "correctAnswerFractions": false, "numRows": "2", "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/"}]}