// Numbas version: exam_results_page_options
{"question_groups": [{"pickingStrategy": "all-ordered", "name": "", "pickQuestions": 1, "questions": [{"name": "Calculate vector magnitude and direction", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Component.gif", "/srv/numbas/media/question-resources/Component.gif"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Let $\\underline{v}=\\left(\\begin{array}{c}\\var{x}\\\\ \\var{y}\\end{array}\\right)$</p>\n<p>You are going to calculate the magnitude and direction of this vector. Refer to the diagram below:</p>\n<p><img src=\"resources/question-resources/Component.gif\"/></p>\n<p></p>", "advice": "", "rulesets": {}, "variables": {"y": {"name": "y", "group": "Ungrouped variables", "definition": "random(1..10 except x)", "description": "", "templateType": "anything"}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "", "templateType": "anything"}, "angle": {"name": "angle", "group": "Ungrouped variables", "definition": "degrees(arctan(y/x))", "description": "", "templateType": "anything"}, "r": {"name": "r", "group": "Ungrouped variables", "definition": "sqrt(x^2+y^2)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["x", "y", "r", "angle"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What is the horizontal component of $\\underline{v}$?</p>", "minValue": "{x}", "maxValue": "{x}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What is the vertical component of $\\underline{v}$?</p>", "minValue": "{y}", "maxValue": "{y}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the magnitude of $\\underline{v}$.</p>", "minValue": "{r}", "maxValue": "{r}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": "0", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the direction of the vector $\\underline{v}$.</p>", "minValue": "{angle}", "maxValue": "{angle}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": 0, "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Calculate component form of vector", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Component.gif", "/srv/numbas/media/question-resources/Component.gif"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>A ship travels $\\var{m}$ nautical miles on a bearing of $\\var{d}^\\circ$.</p>\n<p>You will calculate its horizontal (east/west) component and its vertical (north/south) component.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"northpoint": {"name": "northpoint", "group": "Ungrouped variables", "definition": "if(d<90 or d>270,1,0)", "description": "", "templateType": "anything", "can_override": false}, "eastpoint": {"name": "eastpoint", "group": "Ungrouped variables", "definition": "if(d<180,1,0)", "description": "", "templateType": "anything", "can_override": false}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "m*cos(radians(d))", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "m*sin(radians(d))", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(10..300 #10)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(0..360#5 except [0,45,90,135,180,225,270,315,360])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["y", "x", "m", "d", "northpoint", "eastpoint"], "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>Bearings are measured clockwise from north. Draw a diagram to help you to understand the vector.</p>\n<p>The ship has travelled&nbsp;[[0]] and&nbsp;[[1]].</p>", "gaps": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "dropdownlist", "displayColumns": 0, "showCellAnswerState": true, "choices": ["<p>east</p>", "<p>west</p>"], "matrix": ["{eastpoint}", "1-{eastpoint}"], "distractors": ["", ""]}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "dropdownlist", "displayColumns": 0, "showCellAnswerState": true, "choices": ["<p>north</p>", "<p>south</p>"], "matrix": ["{northpoint}", "1-{northpoint}"], "distractors": ["", ""]}], "sortAnswers": false}, {"type": "information", "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>East and North will give a positive component.</p>\n<p>West and South will give a negative component.</p>"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the horizontal component of the vector.</p>", "stepsPenalty": 0, "steps": [{"type": "information", "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>How far east or west has the ship travelled? (If it is west, then enter a negative number.)</p>\n<p>Refer to Lesson 1 for examples.</p>"}], "minValue": "{x}", "maxValue": "{x}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the vertical component of the vector.</p>", "stepsPenalty": 0, "steps": [{"type": "information", "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>How far north or south has the ship travelled? (If it is south, then enter a negative number.)</p>\n<p>Refer to Lesson 1 for examples.</p>"}], "minValue": "{y}", "maxValue": "{y}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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, "prompt": "<p>Now write the vector in component form.</p>", "correctAnswer": "matrix([{x}],[{y}])", "correctAnswerFractions": false, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": "0", "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": "", "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Arithmetic with vectors", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "variable_groups": [], "ungrouped_variables": ["x1", "y1", "z1", "x2", "y2", "z2", "v1", "v2", "a", "b", "c"], "tags": [], "advice": "", "rulesets": {}, "functions": {}, "variables": {"v1": {"description": "", "name": "v1", "group": "Ungrouped variables", "definition": "matrix([x1],[y1],[z1])", "templateType": "anything"}, "x1": {"description": "", "name": "x1", "group": "Ungrouped variables", "definition": "random(-5..5)", "templateType": "anything"}, "b": {"description": "", "name": "b", "group": "Ungrouped variables", "definition": "random(2..8 except a)", "templateType": "anything"}, "z1": {"description": "", "name": "z1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "templateType": "anything"}, "x2": {"description": "", "name": "x2", "group": "Ungrouped variables", "definition": "random(-5..5)", "templateType": "anything"}, "y2": {"description": "", "name": "y2", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "templateType": "anything"}, "c": {"description": "", "name": "c", "group": "Ungrouped variables", "definition": "random(2..8 except b)", "templateType": "anything"}, "y1": {"description": "", "name": "y1", "group": "Ungrouped variables", "definition": "random(-5..5)", "templateType": "anything"}, "v2": {"description": "", "name": "v2", "group": "Ungrouped variables", "definition": "matrix([x2],[y2],[z2])", "templateType": "anything"}, "a": {"description": "", "name": "a", "group": "Ungrouped variables", "definition": "random(2..5)", "templateType": "anything"}, "z2": {"description": "", "name": "z2", "group": "Ungrouped variables", "definition": "random(-5..5 except z1*y2/y1)", "templateType": "anything"}}, "preamble": {"css": "", "js": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "statement": "<p>Let $\\underline{a}=\\left(\\begin{array}{c}\\var{x1}\\\\ \\var{y1}\\\\ \\var{z1}\\end{array}\\right)$ and let $\\underline{b}=\\left(\\begin{array}{c}\\var{x2}\\\\ \\var{y2}\\\\ \\var{z2}\\end{array}\\right)$</p>\n<p>Calculate the following vectors:</p>", "metadata": {"licence": "None specified", "description": ""}, "parts": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "correctAnswerFractions": false, "marks": "3", "variableReplacements": [], "allowFractions": false, "showCorrectAnswer": true, "markPerCell": true, "numRows": "3", "tolerance": 0, "type": "matrix", "showFeedbackIcon": true, "allowResize": false, "numColumns": 1, "prompt": "<p>What is $\\underline{a}+\\underline{b}$?</p>", "correctAnswer": "{v1+v2}"}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "correctAnswerFractions": false, "marks": "3", "variableReplacements": [], "allowFractions": false, "showCorrectAnswer": true, "markPerCell": true, "numRows": "3", "tolerance": 0, "type": "matrix", "showFeedbackIcon": true, "allowResize": false, "numColumns": 1, "prompt": "<p>What is $\\var{a}\\underline{a}$?</p>", "correctAnswer": "{a}*{v1}"}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "correctAnswerFractions": false, "marks": "3", "variableReplacements": [], "allowFractions": false, "showCorrectAnswer": true, "markPerCell": true, "numRows": "3", "tolerance": 0, "type": "matrix", "showFeedbackIcon": true, "allowResize": false, "numColumns": 1, "prompt": "<p>What is $\\var{b}\\underline{a}-\\var{c}\\underline{b}$?</p>", "correctAnswer": "{b}*{v1}-{c}*{v2}"}], "type": "question"}, {"name": "Colinearity of points", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Let $A,B,C$ be the points</p>\n<p>\\[A=(\\var{x1},\\var{y1},\\var{z1})\\qquad B=(\\var{x2},\\var{y2},\\var{z2})\\qquad C=(\\var{x3},\\var{y3},\\var{z3})\\]</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"z1": {"name": "z1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "x3": {"name": "x3", "group": "Ungrouped variables", "definition": "x2+n*(x2-x1)", "description": "", "templateType": "anything", "can_override": false}, "x2": {"name": "x2", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "can_override": false}, "y2": {"name": "y2", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "y1": {"name": "y1", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "can_override": false}, "z3": {"name": "z3", "group": "Ungrouped variables", "definition": "z2+n*(z2-z1)", "description": "", "templateType": "anything", "can_override": false}, "z2": {"name": "z2", "group": "Ungrouped variables", "definition": "random(-5..5 except z1*y2/y1)", "description": "", "templateType": "anything", "can_override": false}, "y3": {"name": "y3", "group": "Ungrouped variables", "definition": "y2+n*(y2-y1)", "description": "", "templateType": "anything", "can_override": false}, "x1": {"name": "x1", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["x1", "y1", "z1", "x2", "y2", "z2", "n", "x3", "y3", "z3"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What is the vector $\\overrightarrow{AB}$?</p>", "correctAnswer": "matrix([{x2-x1}],[{y2-y1}],[{z2-z1}])", "correctAnswerFractions": false, "numRows": "3", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What is the vector $\\overrightarrow{BC}$?</p>", "correctAnswer": "matrix([{x3-x2}],[{y3-y2}],[{z3-z2}])", "correctAnswerFractions": false, "numRows": "3", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}, {"type": "1_n_2", "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>Are the points colinear?</p>", "stepsPenalty": 0, "steps": [{"type": "information", "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>If&nbsp;$\\overrightarrow{BC}$ is a multiple of $\\overrightarrow{AB}$ then the vectors are parallel and the points are colinear.</p>\n<p>Otherwise the points are not colinear.</p>"}], "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["<p>Yes</p>", "<p>No</p>"], "matrix": ["1", 0], "distractors": ["Correct. $\\overrightarrow{BC}$ is a multiple of $\\overrightarrow{AB}$ so the vectors are parallel", "Incorrect. $\\overrightarrow{BC}$ is a multiple of $\\overrightarrow{AB}$ so the vectors are parallel"]}, {"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>Complete the following:</p>\n<p>$\\overrightarrow{BC}=$ [[0]]$\\times\\overrightarrow{AB}$</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{n}", "maxValue": "{n}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Scalar product of vectors", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Let $\\underline{a}=\\left(\\begin{array}{c}\\var{x1}\\\\ \\var{y1}\\\\ \\var{z1}\\end{array}\\right)$ and let $\\underline{b}=\\left(\\begin{array}{c}\\var{x2}\\\\ \\var{y2}\\\\ \\var{z2}\\end{array}\\right)$</p>", "advice": "", "rulesets": {}, "variables": {"angle": {"name": "angle", "group": "Ungrouped variables", "definition": "degrees(arccos(sp/(r1*r2)))", "description": "", "templateType": "anything"}, "y2": {"name": "y2", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything"}, "x2": {"name": "x2", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "sp": {"name": "sp", "group": "Ungrouped variables", "definition": "x1*x2+y1*y2+z1*z2", "description": "", "templateType": "anything"}, "r2": {"name": "r2", "group": "Ungrouped variables", "definition": "sqrt(x2^2+y2^2+z2^2)", "description": "", "templateType": "anything"}, "x1": {"name": "x1", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "r1": {"name": "r1", "group": "Ungrouped variables", "definition": "sqrt(x1^2+y1^2+z1^2)", "description": "", "templateType": "anything"}, "y1": {"name": "y1", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "z1": {"name": "z1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything"}, "z2": {"name": "z2", "group": "Ungrouped variables", "definition": "random(-5..5 except z1*y2/y1)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["x1", "y1", "z1", "x2", "y2", "z2", "r1", "r2", "sp", "angle"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What is $|\\underline{a}|$?</p>", "stepsPenalty": 0, "steps": [{"type": "information", "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>$|\\underline{a}|$ is the magnitude of vector&nbsp;$\\underline{a}$.&nbsp;Refer to Lesson 1.</p>"}], "minValue": "{r1}", "maxValue": "{r1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What is $|\\underline{b}|$?</p>", "minValue": "{r2}", "maxValue": "{r2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate&nbsp;$\\underline{a}\\cdot\\underline{b}$</p>", "stepsPenalty": 0, "steps": [{"type": "information", "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>$\\underline{a}\\cdot\\underline{b}$ is the scalar product of vectors $\\underline{a}$ and $\\underline{b}$. Refer to Lesson 3.</p>"}], "minValue": "{sp}", "maxValue": "{sp}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Now calculate the angle between&nbsp;vectors $\\underline{a}$ and $\\underline{b}$.</p>", "stepsPenalty": 0, "steps": [{"type": "information", "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>Rearrange the following formula:</p>\n<p>\\[\\underline{a}\\cdot\\underline{b}=|\\underline{a}||\\underline{b}|\\cos{\\theta}\\]</p>\n<p>Refer to Lesson 3 for examples.</p>"}], "minValue": "{angle}", "maxValue": "{angle}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": 0, "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Calculate resultant vector", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Component.gif", "/srv/numbas/media/question-resources/Component.gif"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>A body is subject to two forces:</p>\n<ul>\n<li>$F_1$ acting horizontally with magnitude $\\var{x1}$ N.</li>\n<li>$F_2$ acting at $\\var{d2}^\\circ$ to the horizontal&nbsp;with magnitude $\\var{m2}$ N.</li>\n</ul>\n<p>We will calculate the resultant force $F_r$.</p>\n<p></p>", "advice": "<h4>Part a)</h4>\n<p>$F_1$ is purely horizontal so</p>\n<p>\\[F_1=\\var{mat1}\\]</p>\n<h4>Part b)</h4>\n<p>The horizontal component of $F_2$ is calculated with trigonometry:</p>\n<p>\\[\\var{m2}\\times\\cos(\\var{d2})=\\var{precround(x2,1)}\\]</p>\n<h4>Part c)</h4>\n<p>The vertical component of $F_2$ is calculated with trigonometry:</p>\n<p>\\[\\var{m2}\\times\\sin(\\var{d2})=\\var{precround(y2,1)}\\]</p>\n<h4>Part d)</h4>\n<p>Therefore, in component form:</p>\n<p>\\[F_2=\\var{precround(mat2,1)}\\]</p>\n<h4>Part e)</h4>\n<p>The resultant vector $F_r$ is found by adding the vectors $F_1$ and $F_2$.</p>\n<p>\\[F_r=\\var{mat1}+\\var{precround(mat2,1)}=\\var{precround(mat1+mat2,1)}\\]</p>\n<h4>Part f)</h4>\n<p>The magnitude of the resultant force $F_r$ is calculated with Pythagoras:</p>\n<p>\\[\\sqrt{\\var{precround(x1+x2,1)}^2+\\var{precround(y2,1)}^2}=\\var{precround(mr,0)}\\]</p>\n<h4>Part g)</h4>\n<p>The direction of the resultant force $F_r$ is calculated with trigonometry:</p>\n<p>\\[\\tan^{-1}\\left(\\frac{\\var{precround(y2,1)}}{\\var{precround(x1+x2,1)}}\\right)=\\var{precround(dr,0)}^\\circ\\]</p>\n<p>The resultant force is acting at {precround(dr,0)}&ordm; to the horizontal.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"mat1": {"name": "mat1", "group": "Ungrouped variables", "definition": "matrix([{x1}],[0])", "description": "", "templateType": "anything", "can_override": false}, "mr": {"name": "mr", "group": "Ungrouped variables", "definition": "sqrt((x1+x2)^2+y2^2)", "description": "", "templateType": "anything", "can_override": false}, "mat2": {"name": "mat2", "group": "Ungrouped variables", "definition": "matrix([{x2}],[{y2}])", "description": "", "templateType": "anything", "can_override": false}, "x2": {"name": "x2", "group": "Ungrouped variables", "definition": "m2*cos(radians(d2))", "description": "", "templateType": "anything", "can_override": false}, "y2": {"name": "y2", "group": "Ungrouped variables", "definition": "m2*sin(radians(d2))", "description": "", "templateType": "anything", "can_override": false}, "m2": {"name": "m2", "group": "Ungrouped variables", "definition": "random(10..300 #10)", "description": "", "templateType": "anything", "can_override": false}, "dr": {"name": "dr", "group": "Ungrouped variables", "definition": "degrees(arctan(y2/(x1+x2)))", "description": "", "templateType": "anything", "can_override": false}, "d2": {"name": "d2", "group": "Ungrouped variables", "definition": "random(0..90#5 except [0,45,90])", "description": "", "templateType": "anything", "can_override": false}, "x1": {"name": "x1", "group": "Ungrouped variables", "definition": "random(10..300#10 except m2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["x2", "y2", "d2", "m2", "x1", "mat2", "mat1", "mr", "dr"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "information", "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>Sketch a diagram to help you understand the situation.</p>"}, {"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, "prompt": "<p>Write the vector $F_1$ in component form.</p>", "correctAnswer": "{mat1}", "correctAnswerFractions": false, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": "0", "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the horizontal component of vector $F_2$, in Newtons.</p>", "minValue": "{x2}", "maxValue": "{x2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the vertical component of vector $F_2$, in Newtons.</p>", "minValue": "{y2}", "maxValue": "{y2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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, "prompt": "<p>Now write vector $F_2$ in component form.</p>", "correctAnswer": "{mat2}", "correctAnswerFractions": false, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": "0", "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": "", "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true}, {"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, "prompt": "<p>Calculate the resultant vector $F_r$ of the two vectors $F_1$ and $F_2$.</p>", "stepsPenalty": 0, "steps": [{"type": "information", "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>The resultant of two vectors is calculated by adding them together. Refer to Lesson 2 for examples.</p>"}], "correctAnswer": "{mat1+mat2}", "correctAnswerFractions": false, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": "0", "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": "", "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the magnitude of the resultant force&nbsp;$F_r$, in Newtons.</p>", "minValue": "{mr}", "maxValue": "{mr}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": 0, "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the direction of the resultant force&nbsp;$F_r$, in degrees from horizontal.</p>", "minValue": "{dr}", "maxValue": "{dr}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": 0, "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}], "name": "Maria's copy of Vectors", "percentPass": "75", "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "warn", "message": "<p>5 minutes remaining!</p>"}}, "feedback": {"showanswerstate": true, "advicethreshold": 0, "allowrevealanswer": true, "showtotalmark": true, "feedbackmessages": [], "intro": "", "showactualmark": true}, "duration": 3600, "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "<p>Some basic tasks involving vectors, including converting to/from component form, scalar product, resultant vectors.</p>"}, "showQuestionGroupNames": false, "showstudentname": true, "navigation": {"browse": true, "preventleave": true, "allowregen": true, "showfrontpage": true, "onleave": {"action": "none", "message": ""}, "reverse": true, "showresultspage": "oncompletion"}, "type": "exam", "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}], "extensions": [], "custom_part_types": [], "resources": [["question-resources/Component.gif", "/srv/numbas/media/question-resources/Component.gif"]]}