// Numbas version: finer_feedback_settings
{"name": "LV02 Scalar multiplication of vectors", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "LV02 Scalar multiplication of vectors", "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<h2><strong>Vector Arithmetic</strong></h2>\n<p>Work through the following questions exploring how to multiply a vector by a scalar.</p>\n<p>For the whole of this question:</p>\n<p>$\\bf{a} = \\var{a}$ and $\\bf{b}=\\var{b}$.</p>", "advice": "<p>The vectors in this question have two dimensions but the ideas herein work in any number of dimensions.</p>\n<p>To multiply a vector by a scalar (number) you just multiply each element by that scalar:</p>\n<p>$$<br/>k\\left(\\begin{array}{c}<br/>a \\\\<br/>b \\\\<br/>\\end{array}\\right)&nbsp;=<br/>\\left(\\begin{array}{c}<br/>ak \\\\<br/>bk \\\\<br/>\\end{array}\\right).<br/>$$</p>\n<p>So we have:</p>\n<p>1)</p>\n<p>$$<br/>\\var{m}\\var{a} = \\var{m*a}.<br/>$$</p>\n<p>The second and third part of this question just combine this idea of multiplying a vector by a scalar and the idea that addition and subtraction work by just calculating element by element (as long as all the vectors involved have the same dimensions).&nbsp;&nbsp;</p>\n<p>2)</p>\n<p>$$<br/>\\var{p}\\var{a} + \\var{q}\\var{b} =<br/>\\left(\\begin{array}{c}<br/>\\var{p} \\times \\var{a[0]} + \\var{q} \\times \\var{b[0]} \\\\<br/>\\var{p}\\times \\var{a[1]} + \\var{q} \\times \\var{b[1]}\\\\<br/>\\end{array}\\right) = \\var{p*a+q*b}.<br/>$$</p>\n<p>3)</p>\n<p>$$<br/>\\var{r}\\var{a} - \\var{s}\\var{b} =<br/>\\left(\\begin{array}{c}<br/>\\var{r} \\times \\var{a[0]} - \\var{s} \\times \\var{b[0]} \\\\<br/>\\var{s}\\times \\var{a[1]} - \\var{s} \\times \\var{b[1]}\\\\<br/>\\end{array}\\right) = \\var{r*a-s*b}.<br/>$$</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(-4..4),2))", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "vector(repeat(random(-4..4),2))", "description": "", "templateType": "anything", "can_override": false}, "answeradd": {"name": "answeradd", "group": "Ungrouped variables", "definition": "p*a+q*b", "description": "", "templateType": "anything", "can_override": false}, "answersub": {"name": "answersub", "group": "Ungrouped variables", "definition": "r*a-s*b", "description": "", "templateType": "anything", "can_override": false}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(-9..9 except [-1,0,1,m])", "description": "", "templateType": "anything", "can_override": false}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "random(-9..9 except [-1,0,1,p,m])", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(-9..9 except [-1,0,1])", "description": "", "templateType": "anything", "can_override": false}, "answerma": {"name": "answerma", "group": "Ungrouped variables", "definition": "m*a", "description": "", "templateType": "anything", "can_override": false}, "r": {"name": "r", "group": "Ungrouped variables", "definition": "random(2..9 except [p,m,q])", "description": "", "templateType": "anything", "can_override": false}, "s": {"name": "s", "group": "Ungrouped variables", "definition": "random(2..9 except [p,m,q,r])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "m", "p", "q", "r", "s", "answerma", "answeradd", "answersub"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "matrix", "useCustomName": true, "customName": "1)", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate $\\var{m}\\bf{a}$.</p>", "correctAnswer": "answerma", "correctAnswerFractions": false, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}, {"type": "matrix", "useCustomName": true, "customName": "2)", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate $\\var{p}\\bf{a}+\\var{q}\\bf{b}$.</p>", "correctAnswer": "answeradd", "correctAnswerFractions": false, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}, {"type": "matrix", "useCustomName": true, "customName": "3)", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate $\\var{r}\\bf{a} - 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