// Numbas version: exam_results_page_options
{"name": "Vectors Introduction", "metadata": {"description": "<p>easy vector addition and scalar multiplication, for practice after Section 1 of lectures.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "duration": 0, "percentPass": "0", "showQuestionGroupNames": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": "1", "questionNames": ["", "", "", ""], "questions": [{"name": "Adding vectors", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Tatiana Tyukina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/341/"}, {"name": "Julia Goedecke", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5121/"}], "tags": [], "metadata": {"description": "<p>Adding vectors of random size. Advice (i.e. solution) has conditional visibility to show only the correct size.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "advice": "<p>We can add any two vectors which have the same size. To add the vectors, we add each entry separately.</p>\n<p><span data-jme-visible=\"rows=2\">\\[\\var{v}+\\var{w}=\\begin{pmatrix}\\simplify[]{{v[0]}+{w[0]}} \\\\ \\simplify[]{{v[1]}+{w[1]}}&nbsp; \\end{pmatrix} = \\var{vectorsum}\\]</span></p>\n<p><span data-jme-visible=\"rows=3\">\\[\\var{v}+\\var{w}=\\begin{pmatrix}\\simplify[]{{v[0]}+{w[0]}} \\\\ \\simplify[]{{v[1]}+{w[1]}} \\\\\\simplify[]{{v[2]}+{w[2]}}&nbsp; \\end{pmatrix} = \\var{vectorsum}\\]</span></p>\n<p><span data-jme-visible=\"rows=4\">\\[\\var{v}+\\var{w}=\\begin{pmatrix}\\simplify[]{{v[0]}+{w[0]}} \\\\ \\simplify[]{{v[1]}+{w[1]}} \\\\\\simplify[]{{v[2]}+{w[2]}} \\\\\\simplify[]{{v[3]}+{w[3]}} \\end{pmatrix} = \\var{vectorsum}\\]</span></p>\n<p><span data-jme-visible=\"rows=5\">\\[\\var{v}+\\var{w}=\\begin{pmatrix}\\simplify[]{{v[0]}+{w[0]}} \\\\ \\simplify[]{{v[1]}+{w[1]}} \\\\\\simplify[]{{v[2]}+{w[2]}} \\\\\\simplify[]{{v[3]}+{w[3]}} \\\\ \\simplify[]{{v[4]}+{w[4]}} \\end{pmatrix} = \\var{vectorsum}\\]</span></p>\n<p><span data-jme-visible=\"rows=6\">\\[\\var{v}+\\var{w}=\\begin{pmatrix}\\simplify[]{{v[0]}+{w[0]}} \\\\ \\simplify[]{{v[1]}+{w[1]}} \\\\\\simplify[]{{v[2]}+{w[2]}} \\\\\\simplify[]{{v[3]}+{w[3]}} \\\\ \\simplify[]{{v[4]}+{w[4]}} \\\\\\simplify[]{{v[5]}+{w[5]}} \\end{pmatrix} = \\var{vectorsum}\\]</span></p>", "rulesets": {}, "variables": {"vectorsum": {"name": "vectorsum", "group": "Ungrouped variables", "definition": "v+w", "description": "<p>Sum of vectors v and w</p>", "templateType": "anything"}, "rows": {"name": "rows", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "<p>size of the vectors</p>", "templateType": "anything"}, "v": {"name": "v", "group": "Ungrouped variables", "definition": "vector((repeat(random(-5..5),rows)))", "description": "<p>randomly generated vector of random size.</p>", "templateType": "anything"}, "w": {"name": "w", "group": "Ungrouped variables", "definition": "vector(repeat(random(-5..5),rows))", "description": "<p>randomly generated matrix of same size as matrixA</p>", "templateType": "anything"}}, "variablesTest": {"condition": "not is_zero(v) and not is_zero(w)", "maxRuns": 100}, "ungrouped_variables": ["vectorsum", "rows", "v", "w"], "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>Calculate \\(\\var{v} + \\var{w}=\\) [[0]].</p>\n<p></p>", "gaps": [{"type": "matrix", "useCustomName": true, "customName": "Gap 0", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "vectorsum", "correctAnswerFractions": false, "numRows": "1", "numColumns": "1", "allowResize": true, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": "1", "minRows": 1, "maxRows": 0}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Adding and subtracting vectors", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Tatiana Tyukina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/341/"}, {"name": "Julia Goedecke", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5121/"}], "tags": [], "metadata": {"description": "<p>Adding and subtracting vectors of random size, including resolving brackets. Advice (i.e. solution) has conditional visibility to show only the correct size.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Let \\(u=\\var{u}\\), \\(v=\\var{v}\\) and \\(w=\\var{w}\\).</p>", "advice": "<p>We add and subtract vectors entry by entry.</p>\n<p><strong>Part a)</strong></p>\n<p><span data-jme-visible=\"rows=2\">\\[\\simplify[unitFactor]{v+{sign1}w}=\\begin{pmatrix}\\simplify[]{{v[0]}+{sign1*w[0]}} \\\\ \\simplify[]{{v[1]}+{sign1*w[1]}}&nbsp; \\end{pmatrix} = \\var{v+sign1*w}\\]</span></p>\n<p><span data-jme-visible=\"rows=3\">\\[\\simplify[unitFactor]{v+{sign1}w}=\\begin{pmatrix}\\simplify[]{{v[0]}+{sign1*w[0]}} \\\\ \\simplify[]{{v[1]}+{sign1*w[1]}} \\\\\\simplify[]{{v[2]}+{sign1*w[2]}}&nbsp; \\end{pmatrix} = \\var{v+sign1*w}\\]</span></p>\n<p><span data-jme-visible=\"rows=4\">\\[\\simplify[unitFactor]{v+{sign1}w}=\\begin{pmatrix}\\simplify[]{{v[0]}+{sign1*w[0]}} \\\\ \\simplify[]{{v[1]}+{sign1*w[1]}} \\\\\\simplify[]{{v[2]}+{sign1*w[2]}} \\\\\\simplify[]{{v[3]}+{sign1*w[3]}} \\end{pmatrix} = \\var{v+sign1*w}\\] <br/></span></p>\n<p><strong>Part b)</strong></p>\n<p><span data-jme-visible=\"rows=2\">\\[v -(\\simplify[unitFactor,!expandBrackets]{u+{sign2}*w})=\\begin{pmatrix}\\simplify[]{{v[0]}-({u[0]}+{sign2*w[0]})} \\\\ \\simplify[]{{v[1]}-({u[1]}+{sign2*w[1]})} \\end{pmatrix}=\\var{v-(u+sign2*w)} \\]</span></p>\n<p><span data-jme-visible=\"rows=3\">\\[v -(\\simplify[unitFactor,!expandBrackets]{u+{sign2}*w})=\\begin{pmatrix}\\simplify[]{{v[0]}-({u[0]}+{sign2*w[0]})} \\\\ \\simplify[]{{v[1]}-({u[1]}+{sign2*w[1]})} \\\\\\simplify[]{{v[2]}-({u[2]}+{sign2*w[2]})} \\end{pmatrix}=\\var{v-(u+sign2*w)} \\]</span></p>\n<p><span data-jme-visible=\"rows=4\">\\[v -(\\simplify[unitFactor,!expandBrackets]{u+{sign2}*w})=\\begin{pmatrix}\\simplify[]{{v[0]}-({u[0]}+{sign2*w[0]})} \\\\ \\simplify[]{{v[1]}-({u[1]}+{sign2*w[1]})} \\\\\\simplify[]{{v[2]}-({u[2]}+{sign2*w[2]})} \\\\\\simplify[]{{v[3]}-({u[3]}+{sign2*w[3]})} \\end{pmatrix}=\\var{v-(u+sign2*w)} \\]</span></p>", "rulesets": {}, "variables": {"rows": {"name": "rows", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "<p>size of the vectors</p>", "templateType": "anything"}, "v": {"name": "v", "group": "Ungrouped variables", "definition": "vector((repeat(random(-5..5),rows)))", "description": "<p>randomly generated vector of random size.</p>", "templateType": "anything"}, "w": {"name": "w", "group": "Ungrouped variables", "definition": "vector(repeat(random(-5..5),rows))", "description": "<p>randomly generated vector of same size as v</p>", "templateType": "anything"}, "u": {"name": "u", "group": "Ungrouped variables", "definition": "vector(repeat(random(-5..5),rows))", "description": "<p>randomly generated vector of same size as v</p>", "templateType": "anything"}, "sign1": {"name": "sign1", "group": "Ungrouped variables", "definition": "random(-1..1 except 0)", "description": "<p>randomly generated sign</p>", "templateType": "anything"}, "sign2": {"name": "sign2", "group": "Ungrouped variables", "definition": "random(-1..1 except 0)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "not is_zero(v) and not is_zero(w) and not is_zero(u)", "maxRuns": 100}, "ungrouped_variables": ["rows", "v", "w", "u", "sign1", "sign2"], "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>Calculate \\(\\simplify{v + {sign1}*w}=\\) [[0]].</p>\n<p></p>", "gaps": [{"type": "matrix", "useCustomName": true, "customName": "Gap 0", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "v+sign1*w", "correctAnswerFractions": false, "numRows": "1", "numColumns": "1", "allowResize": true, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": "1", "minRows": 1, "maxRows": 0}], "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>Calculate \\(v -(\\simplify[unitFactor,!expandBrackets]{u+{sign2}*w})=\\) [[0]].</p>\n<p></p>", "gaps": [{"type": "matrix", "useCustomName": true, "customName": "Gap 0", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "v-(u+sign2*w)", "correctAnswerFractions": false, "numRows": "1", "numColumns": "1", "allowResize": true, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": "1", "minRows": 1, "maxRows": 0}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Calculating with vectors", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Tatiana Tyukina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/341/"}, {"name": "Julia Goedecke", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5121/"}], "tags": [], "metadata": {"description": "<p>Calculating with vectors of random size, including resolving brackets. Advice (i.e. solution) has conditional visibility to show only the correct size.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Let \\(u=\\var{u}\\), \\(v=\\var{v}\\) and \\(w=\\var{w}\\).</p>", "advice": "<p>To multiply a vector by a scalar, we multiply each entry by that scalar. To add the resulting vectors, we add them entry by entry. Don't forget to resolve the brackets.</p>\n<p><strong>Part a)</strong></p>\n<p><span data-jme-visible=\"rows=2\">\\[\\simplify{v + {lambda}*(u-{mu}w)}=\\begin{pmatrix}\\simplify[]{{v[0]}+{lambda}*({u[0]}-{mu}*{w[0]})} \\\\&nbsp;\\simplify[]{{v[1]}+{lambda}*({u[1]}-{mu}*{w[1]})} \\end{pmatrix} = \\var{v+lambda*(u-mu*w)}\\]</span></p>\n<p><span data-jme-visible=\"rows=3\">\\[\\simplify{v + {lambda}*(u-{mu}w)}=\\begin{pmatrix}\\simplify[]{{v[0]}+{lambda}*({u[0]}-{mu}*{w[0]})} \\\\ \\simplify[]{{v[1]}+{lambda}*({u[1]}-{mu}*{w[1]})} \\\\\\simplify[]{{v[2]}+{lambda}*({u[2]}-{mu}*{w[2]})} \\end{pmatrix} = \\var{v+lambda*(u-mu*w)}\\]</span></p>\n<p><span data-jme-visible=\"rows=4\">\\[\\simplify{v + {lambda}*(u-{mu}w)}=\\begin{pmatrix}\\simplify[]{{v[0]}+{lambda}*({u[0]}-{mu}*{w[0]})} \\\\ \\simplify[]{{v[1]}+{lambda}*({u[1]}-{mu}*{w[1]})} \\\\\\simplify[]{{v[2]}+{lambda}*({u[2]}-{mu}*{w[2]})} \\\\\\simplify[]{{v[3]}+{lambda}*({u[3]}-{mu}*{w[3]})} \\end{pmatrix} = \\var{v+lambda*(u-mu*w)}\\] <br/></span></p>\n<p><strong>Part b)</strong></p>\n<p><span data-jme-visible=\"rows=2\">\\[\\simplify[]{{lambda1}*u +{mu}({mu1}v+{lambda}w)}=\\begin{pmatrix}\\simplify[]{{lambda1}{u[0]}+{mu}({mu1}{v[0]}+{lambda}*{w[0]})} \\\\\\simplify[]{{lambda1}{u[1]}+{mu}({mu1}{v[1]}+{lambda}*{w[1]})}\\end{pmatrix}=\\var{lambda1*u+mu*(mu1*v+lambda*w)} \\]</span></p>\n<p><span data-jme-visible=\"rows=3\">\\[\\simplify[]{{lambda1}*u +{mu}({mu1}v+{lambda}w)}=\\begin{pmatrix}\\simplify[]{{lambda1}{u[0]}+{mu}({mu1}{v[0]}+{lambda}*{w[0]})}\\\\ \\simplify[]{{lambda1}{u[1]}+{mu}({mu1}{v[1]}+{lambda}*{w[1]})}\\\\ \\simplify[]{{lambda1}{u[2]}+{mu}({mu1}{v[2]}+{lambda}*{w[2]})}\\end{pmatrix}=\\var{lambda1*u+mu*(mu1*v+lambda*w)} \\]</span></p>\n<p><span data-jme-visible=\"rows=4\">\\[\\simplify[]{{lambda1}*u +{mu}({mu1}v+{lambda}w)}=\\begin{pmatrix}\\simplify[]{{lambda1}{u[0]}+{mu}({mu1}{v[0]}+{lambda}*{w[0]})}\\\\ \\simplify[]{{lambda1}{u[1]}+{mu}({mu1}{v[1]}+{lambda}*{w[1]})}\\\\&nbsp;\\simplify[]{{lambda1}{u[2]}+{mu}({mu1}{v[2]}+{lambda}*{w[2]})} \\\\&nbsp;\\simplify[]{{lambda1}{u[3]}+{mu}({mu1}{v[3]}+{lambda}*{w[3]})} \\end{pmatrix}=\\var{lambda1*u+mu*(mu1*v+lambda*w)} \\]</span></p>", "rulesets": {}, "variables": {"rows": {"name": "rows", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "<p>size of the vectors</p>", "templateType": "anything"}, "v": {"name": "v", "group": "Ungrouped variables", "definition": "vector((repeat(random(-5..5),rows)))", "description": "<p>randomly generated vector of random size.</p>", "templateType": "anything"}, "w": {"name": "w", "group": "Ungrouped variables", "definition": "vector(repeat(random(-5..5),rows))", "description": "<p>randomly generated vector of same size as v</p>", "templateType": "anything"}, "u": {"name": "u", "group": "Ungrouped variables", "definition": "vector(repeat(random(-5..5),rows))", "description": "<p>randomly generated vector of same size as v</p>", "templateType": "anything"}, "sign1": {"name": "sign1", "group": "Ungrouped variables", "definition": "random(-1..1 except 0)", "description": "<p>randomly generated sign</p>", "templateType": "anything"}, "sign2": {"name": "sign2", "group": "Ungrouped variables", "definition": "random(-1..1 except 0)", "description": "", "templateType": "anything"}, "lambda": {"name": "lambda", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "templateType": "anything"}, "mu": {"name": "mu", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "templateType": "anything"}, "lambda1": {"name": "lambda1", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything"}, "mu1": {"name": "mu1", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "not is_zero(v) and not is_zero(w) and not is_zero(u) and lambda<>lambda1 and mu<>mu1 and mu1<>lambda and mu<>lambda and mu<>lambda1 and lambda1<>mu1", "maxRuns": 100}, "ungrouped_variables": ["rows", "v", "w", "u", "sign1", "sign2", "lambda", "mu", "lambda1", "mu1"], "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>Calculate \\(\\simplify{v + {lambda}*(u-{mu}w)}=\\) [[0]].</p>\n<p></p>", "gaps": [{"type": "matrix", "useCustomName": true, "customName": "Gap 0", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "matrix", "useCustomName": true, "customName": "wrong brackets", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "<p>You seem to have forgotten to multiply out the bracket properly.</p>", "useAlternativeFeedback": false, "correctAnswer": "v+lambda*u-mu*w", "correctAnswerFractions": false, "numRows": "1", "numColumns": "1", "allowResize": true, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": "1", "minRows": 1, "maxRows": 0}], "correctAnswer": "v+lambda*(u-mu*w)", "correctAnswerFractions": false, "numRows": "1", "numColumns": "1", "allowResize": true, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": "1", "minRows": 1, "maxRows": 0}], "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>Calculate \\(\\simplify[]{{lambda1}*u +{mu}({mu1}v+{lambda}w)}=\\) [[0]].</p>\n<p></p>", "gaps": [{"type": "matrix", "useCustomName": true, "customName": "Gap 0", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "matrix", "useCustomName": true, "customName": "wrong brackets", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "<p>You seem to have forgotten to multiply out the bracket properly.</p>", "useAlternativeFeedback": false, "correctAnswer": "lambda1*u +mu*mu1*v+lambda*w", "correctAnswerFractions": false, "numRows": "1", "numColumns": "1", "allowResize": true, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": "1", "minRows": 1, "maxRows": 0}], "correctAnswer": "lambda1*u +mu*(mu1*v+lambda*w)", "correctAnswerFractions": false, "numRows": "1", "numColumns": "1", "allowResize": true, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": "1", "minRows": 1, "maxRows": 0}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Vectors - Introductory exercise (WB Q1.1 randomised)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julia Goedecke", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5121/"}], "tags": [], "metadata": {"description": "<p>Simple vector addition and scalar multiplication in \\(\\mathbb{R}^2\\).</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Calculate the following:</p>", "advice": "<p>We add vectors entry by entry, and we multiply a vector by a scalar by multiplying each entry.</p>\n<p><strong>Part a) </strong>Zero times any vector is the zero vector: \\(0\\cdot \\begin{pmatrix}x_1\\\\x_2\\end{pmatrix}=\\begin{pmatrix}0\\\\0\\end{pmatrix}\\).</p>\n<p><strong>Part b) </strong>\\[\\var{v1}+\\var{v2}=\\begin{pmatrix}\\simplify[]{{v1[0]}+{v2[0]}} \\\\ \\simplify[]{{v1[1]}+{v2[1]}}&nbsp; \\end{pmatrix} = \\var{v1+v2}\\]</p>\n<p><strong>Part c) </strong>\\[\\var{mu1}\\begin{pmatrix}\\sqrt{\\var{n}}\\\\\\var{w1[1]}\\end{pmatrix}+\\var{mu2}\\begin{pmatrix}-\\frac{\\sqrt{\\var{n}}}{\\var{b}}\\\\\\var{w2[1]}\\end{pmatrix}=\\begin{pmatrix}\\var{mu1}\\sqrt{\\var{n}}-\\var{mu2}\\frac{\\sqrt{\\var{n}}}{\\var{b}}\\\\\\var{mu1}\\var{w1[1]}+\\var{mu2}\\var{w2[1]}\\end{pmatrix} = \\var{mu1*w1+mu2*w2}\\]</p>", "rulesets": {}, "variables": {"v1": {"name": "v1", "group": "Ungrouped variables", "definition": "vector((repeat(random(1..5),2)))", "description": "", "templateType": "anything"}, "v2": {"name": "v2", "group": "Ungrouped variables", "definition": "vector((repeat(random(1..5),2)))", "description": "", "templateType": "anything"}, "w1": {"name": "w1", "group": "Ungrouped variables", "definition": "vector(sqrt(n),pi)", "description": "", "templateType": "anything"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(2..5 except 4)", "description": "", "templateType": "anything"}, "w2": {"name": "w2", "group": "Ungrouped variables", "definition": "vector(-w1[0]/b,pi)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..3)", "description": "", "templateType": "anything"}, "mu1": {"name": "mu1", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "templateType": "anything"}, "mu2": {"name": "mu2", "group": "Ungrouped variables", "definition": "b*mu1", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "v1<>v2", "maxRuns": 100}, "ungrouped_variables": ["v1", "v2", "w1", "w2", "n", "b", "mu1", "mu2"], "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": "<table>\n<tbody>\n<tr>\n<td rowspan=\"2\">\\(0\\cdot \\begin{pmatrix}x_1\\\\x_2\\end{pmatrix}= \\left( \\begin{matrix}\\phantom{.}\\\\\\phantom{.}\\\\\\phantom{.}\\end{matrix}\\right.\\)</td>\n<td>[[0]]</td>\n<td rowspan=\"2\">\\(\\left.\\begin{matrix}\\phantom{.}\\\\\\phantom{.}\\\\\\phantom{.}\\end{matrix}\\right)\\)</td>\n</tr>\n<tr>\n<td>[[1]]</td>\n</tr>\n</tbody>\n</table>\n<p></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "0", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "0", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}], "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 style=\"text-align: center;\">\\(\\var{v1}+\\var{v2}=\\) [[0]].</p>", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "v1+v2", "correctAnswerFractions": false, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "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>Enter \\(\\pi\\) as pi.</p>\n<table>\n<tbody>\n<tr>\n<td rowspan=\"2\">\\(\\var{mu1}\\begin{pmatrix}\\sqrt{\\var{n}}\\\\\\var{w1[1]}\\end{pmatrix}+\\var{mu2}\\begin{pmatrix}-\\frac{\\sqrt{\\var{n}}}{\\var{b}}\\\\\\var{w2[1]}\\end{pmatrix}= \\left( \\begin{matrix}\\phantom{.}\\\\\\phantom{.}\\\\\\phantom{.}\\end{matrix}\\right.\\)</td>\n<td>[[0]]</td>\n<td rowspan=\"2\">\\(\\left.\\begin{matrix}\\phantom{.}\\\\\\phantom{.}\\\\\\phantom{.}\\end{matrix}\\right)\\)</td>\n</tr>\n<tr>\n<td>[[1]]</td>\n</tr>\n</tbody>\n</table>\n<p></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "0", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{(mu1+mu2)}*pi", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}], "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": false, "showresultspage": "oncompletion", "navigatemode": "menu", "onleave": {"action": "none", "message": ""}, "preventleave": true, "startpassword": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "<p>Here are a few questions for you to practice calculating with vectors. For each question, you can \"try another question like this\", which will give you different numbers, and different sizes of vectors! We have only done vectors with two or three entries so far, but you should be able to extrapolate to more entries yourself. If you want to try one with two or three entries first, just press \"try another question like this\" until you get the size you want.</p>", "reviewshowscore": true, "reviewshowfeedback": true, "reviewshowexpectedanswer": true, "reviewshowadvice": true, "feedbackmessages": []}, "contributors": [{"name": "Julia Goedecke", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5121/"}], "extensions": [], "custom_part_types": [], "resources": []}