// Numbas version: finer_feedback_settings
{"name": "LV03 Vectors and geometry", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "LV03 Vectors and geometry", "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>This question is about understanding the use of vectors to prove that a shape is a trapezium.</p>\n<p>The quadrilateral ABCD is shown below.</p>\n<p>$$<br/>A = (\\var{a[0]},\\var{a[1]})\\\\<br/>B = (\\var{b[0]},\\var{b[1]})\\\\<br/>C = (\\var{c[0]},\\var{c[1]})\\\\<br/>D = (\\var{d[0]},\\var{d[1]})<br/>$$</p>\n<p>{geogebra_applet('https://www.geogebra.org/m/xb9bvcaa',defs)}</p>", "advice": "<p><strong>Vectors between points</strong></p>\n<p>In order to undertstand how to find the vector between two points it is helpful to know what a \"position vector\" and \"direction vector\" are. This advice should cover parts a), b), e) and f) of this question.</p>\n<p>A position vector is defined as a vector that symbolises the location of any given point with respect to the origin. It can be thought of as a coordinate point, but written as a column vector - top entry is the \"x-coordinate\" and the bottome entry is the \"y-coordinate\". For example:</p>\n<p>The point $B$ has coordinates $(\\var{B[0]},\\var{B[1]})$ and it has position vector, denoted $\\bf{b}$, given as $\\bf{b} = \\var{b}$.</p>\n<p>A direction vector is defined as a vector that symbolises a direction and a distance in that direction but with no specified \"starting point\". In 2D it can be summarized as an instruction to go the top element number of units left or right based on the sign of the element and the bottom element number of units up or down based on the sign of the element.</p>\n<p>So the direction vector from $B$ to $C$ can be worked out by looking at a route from $B$ to $C$ that travels along the position vectors given. Starting at $B$ we have to go <strong>backwards&nbsp;</strong>down $\\bf{b}$ to the origin and then <strong>forwards&nbsp;</strong>along $\\bf{c}$. This corresponds to doing \"minus\" $\\bf{b}$ and \"positive\" $\\bf{c}$:</p>\n<p>$$<br/>\\vec{BC} = (-)\\bf{b} + \\bf{c} = \\bf{c}-\\bf{b} = \\var{c}-\\var{b} = \\var{c-b}.<br/>$$</p>\n<p><strong>Parallel vectors</strong></p>\n<p>If one vector is a multiple of another then they are vectors that point in the same direction. This means they are parallel. You just need to check what the multplier is between corresponding elements in each vector. If it is the same for both pairs of elements then the vectors are parallel (and if not then they are not).</p>\n<p>For example, $\\vec{BC} = \\var{bc}$ and $\\vec{AD} = \\var{k*BC}.$ Since $\\frac{\\var{k*BC[0]}}{\\var{BC[0]}} = \\var{k}$ which gives the same multiplier as $\\frac{\\var{k*BC[1]}}{\\var{BC[1]}} = \\var{k}$ then $\\vec{BC}$ and $\\vec{AD}$ are parallel.</p>\n<p><strong>Conclusions about shapes</strong></p>\n<p>This question is looking at a trapezium specifically. The key properties of a trapezium are that it is a quadrilateral and there is one pair of parallel sides. This question goes through establishing that one pair of sides are parallel and then does the calculations to show that the other pair is not parallel. At this point we can conclude that $ABCD$ is a trapezium.</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": ["geogebra"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"A": {"name": "A", "group": "Ungrouped variables", "definition": "vector(0,0)", "description": "", "templateType": "anything", "can_override": false}, "AB": {"name": "AB", "group": "Ungrouped variables", "definition": "vector(0,random(2..7))", "description": "", "templateType": "anything", "can_override": false}, "B": {"name": "B", "group": "Ungrouped variables", "definition": "A+ab", "description": "", "templateType": "anything", "can_override": false}, "BC": {"name": "BC", "group": "Ungrouped variables", "definition": "vector(repeat(random(2..5),2))", "description": "", "templateType": "anything", "can_override": false}, "C": {"name": "C", "group": "Ungrouped variables", "definition": "B+bc", "description": "", "templateType": "anything", "can_override": false}, "D": {"name": "D", "group": "Ungrouped variables", "definition": "A+k*bc", "description": "", "templateType": "anything", "can_override": false}, "defs": {"name": "defs", "group": "Ungrouped variables", "definition": "[\n  ['A',A],\n  ['B',B],\n  ['C',C],\n  ['D',D]\n]", "description": "", "templateType": "anything", "can_override": false}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["A", "AB", "B", "BC", "C", "D", "defs", "k"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "matrix", "useCustomName": true, "customName": "a)", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the vector $\\vec{BC}$.</p>", "correctAnswer": "-b+c", "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": "b)", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the vector $\\vec{AD}$.</p>", "correctAnswer": "-a+d", "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": true, "customName": "c)", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Hence, it is possible to write $\\vec{AD} = k \\times \\vec{BC}$. Find the value of $k$.</p>", "minValue": "k", "maxValue": "k", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "1_n_2", "useCustomName": true, "customName": "d)", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The fact that $\\vec{AD}$ can be written as a multiplier times by $\\vec{BC}$ means:&nbsp;</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["The sides $BC$ and $AD$ are parallel.", "The sides $BC$ and $AD$ are perpendicular.", "The points $A$, $B$, $C$, and $D$ are colinear.", "Shape $ABCD$ is a trapezium."], "matrix": ["1", 0, 0, 0], "distractors": ["", "", "", ""]}, {"type": "matrix", "useCustomName": true, "customName": "e)", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the vector $\\vec{AB}$.</p>", "correctAnswer": "ab", "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": "f)", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the vector $\\vec{CD}$.</p>", "correctAnswer": "-c+d", "correctAnswerFractions": false, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}, {"type": "m_n_2", "useCustomName": true, "customName": "g)", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Up to this point in the question we have now shown which of the following:</p>", "minMarks": 0, "maxMarks": "1", "shuffleChoices": true, "displayType": "checkbox", "displayColumns": 0, "minAnswers": "0", "maxAnswers": "0", "warningType": "none", "showCellAnswerState": true, "markingMethod": "all-or-nothing", "choices": ["Sides $BC$ and $AD$ are parallel.", "Sides $AB$ and $CD$ are not parallel.", "The shape $ABCD$ is a trapezium."], "matrix": ["1", "1", "1"], "distractors": ["", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "resources": []}]}], "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}]}