// Numbas version: exam_results_page_options
{"name": "My First Exam", "metadata": {"description": "<p>A couple of questions. Some hard, some easy.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", ""], "variable_overrides": [[], [], [], []], "questions": [{"name": "Subtraction", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "tags": ["algorithms", "minus", "subtracting", "subtraction", "take away"], "metadata": {"description": "<p>The subtraction algortihm using the borrow and pay back method with integers.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Write the following question down on paper and evaluate&nbsp;it without using a calculator.</p>\n<p>If you are unsure of how to do&nbsp;a question, click on <em>Show steps</em> to see the full working. Then, once you understand how to do the question, click on <em>Try another question like this one</em> to start again.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"newtopten": {"name": "newtopten", "group": "c", "definition": "if(unitdiff>=0,top[1],top[1]-1)", "description": "", "templateType": "anything", "can_override": false}, "ansunit": {"name": "ansunit", "group": "c", "definition": "mod(ans,10)", "description": "", "templateType": "anything", "can_override": false}, "hundiff": {"name": "hundiff", "group": "c", "definition": "if(tendiff>=0,top[2]-bot[2],top[2]-1-bot[2])", "description": "", "templateType": "anything", "can_override": false}, "botnum": {"name": "botnum", "group": "c", "definition": "bot[0]+bot[1]*10+bot[2]*100", "description": "", "templateType": "anything", "can_override": false}, "ansten": {"name": "ansten", "group": "c", "definition": "mod(floor(ans/10),10)", "description": "", "templateType": "anything", "can_override": false}, "tendiff": {"name": "tendiff", "group": "c", "definition": "if(unitdiff>=0,top[1]-bot[1],top[1]-1-bot[1])", "description": "", "templateType": "anything", "can_override": false}, "unitdiff": {"name": "unitdiff", "group": "c", "definition": "top[0]-bot[0]", "description": "", "templateType": "anything", "can_override": false}, "bot": {"name": "bot", "group": "c", "definition": "random(\nif(top[0]<9,[random(top[0]+1..9), random(top[1]..9), random(1..top[2]-1)],\n      if(top[1]<9,[random(top[0]..9), random(top[1]+1..9), random(1..top[2]-1)],\"error\")),\nif(top[1]<9,[random(0..top[0]), random(top[1]+1..9), random(1..top[2]-1)],\n      if(top[1]=9,[random(top[0]..9), random(0..9), random(1..top[2]-1)],\"error\"))\n)\n", "description": "<p>This should force some borrowing and <g class=\"gr_ gr_4 gr-alert gr_gramm gr_inline_cards gr_run_anim Grammar multiReplace\" id=\"4\" data-gr-id=\"4\">paying</g> back, and that the&nbsp;final answer&nbsp;is positive.</p>", "templateType": "anything", "can_override": false}, "newtophun": {"name": "newtophun", "group": "c", "definition": "if(tendiff>=0,top[2],top[2]-1)", "description": "", "templateType": "anything", "can_override": false}, "topnum": {"name": "topnum", "group": "c", "definition": "top[0]+top[1]*10+top[2]*100", "description": "", "templateType": "anything", "can_override": false}, "anshun": {"name": "anshun", "group": "c", "definition": "mod(floor(ans/100),10)", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "c", "definition": "topnum-botnum", "description": "", "templateType": "anything", "can_override": false}, "top": {"name": "top", "group": "c", "definition": "random([random(0..9),random(1..8),random(2..9)],[random(0..8),random(1..9),random(2..9)])", "description": "<p>Borrowing from the hundreds to do the units is not covered with this randomisation. We will do that in another part.</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "c", "variables": ["top", "bot", "topnum", "botnum", "ans", "unitdiff", "tendiff", "hundiff", "ansunit", "ansten", "anshun", "newtopten", "newtophun"]}], "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>$\\var{topnum}-\\var{botnum} = $&nbsp;[[0]]</p>", "stepsPenalty": "1", "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>Generally&nbsp;we set up $\\var{topnum}-\\var{botnum}$&nbsp;with the <em>ones,&nbsp;</em><em>tens</em> and&nbsp;<em>hundreds</em>&nbsp;columns lined up vertically:</p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td>$\\var{top[2]}$</td>\n<td>$\\var{top[1]}$</td>\n<td>$\\var{top[0]}$</td>\n<td>$-$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\var{bot[2]}$</td>\n<td>$\\var{bot[1]}$</td>\n<td>$\\var{bot[0]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td></td>\n<td>$\\phantom{0}$</td>\n<td></td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>Now&nbsp;we try to subtract the digits in the <g class=\"gr_ gr_38 gr-alert gr_gramm gr_inline_cards gr_run_anim Grammar multiReplace\" id=\"38\" data-gr-id=\"38\">ones</g> column.</p>\n<p><span data-jme-visible=\"unitdiff&gt;=0\">Since this is $\\var{ansunit}$ we&nbsp;write $\\var{ansunit}$ under the line in the ones column.</span></p>\n<p><span data-jme-visible=\"unitdiff&lt;0\">Since we can't&nbsp;take $\\var{bot[0]}$ away from $\\var{top[0]}$ (without using negative numbers) we <em>borrow</em>&nbsp;a ten from the tens column. This means we cross out the $\\var{top[1]}$ in the tens column&nbsp;and replace it with a $\\var{top[1]-1}$, and the $\\var{top[0]}$ becomes a $\\var{10+top[0]}$. Now we can do&nbsp;$\\var{10+top[0]}-\\var{bot[0]}$, and write the result, $\\var{ansunit}$, under the line in the ones column.</span>&nbsp;</p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td>$\\overset{\\phantom{1}}{\\var{top[2]}}$</td>\n<td><span data-jme-visible=\"unitdiff&lt;0\">$\\overset{\\color{red}{\\var{newtopten}}}{\\var{top[1]}\\mkern-7.5mu\\color{red}/}$</span> <span data-jme-visible=\"unitdiff&gt;=0\">$\\overset{\\phantom{1}}{\\var{top[1]}}$</span></td>\n<td><span data-jme-visible=\"unitdiff&lt;0\">$\\color{red}{^1}\\overset{\\phantom{1}}{\\var{top[0]}}$</span> <span data-jme-visible=\"unitdiff&gt;=0\">$\\overset{\\phantom{1}}{\\var{top[0]}}$</span></td>\n<td>$-$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\var{bot[2]}$</td>\n<td>$\\var{bot[1]}$</td>\n<td>$\\var{bot[0]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td></td>\n<td></td>\n<td>$\\color{red}{\\var{ansunit}}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>Now we&nbsp;try to subtract the digits in the tens column.</p>\n<p><span data-jme-visible=\"tendiff&gt;=0\">Since this is $\\var{ansten}$ we&nbsp;write $\\var{ansten}$ under the line in the tens column.</span></p>\n<p><span data-jme-visible=\"tendiff&lt;0\">Since we can't&nbsp;take $\\var{bot[1]}$ away from $\\var{newtopten}$ (without using negative numbers) we <em>borrow</em>&nbsp;a hundred from the hundred column. This means we cross out the $\\var{top[2]}$ in the hundreds column&nbsp;and replace it with a $\\var{top[2]-1}$, and the $\\var{newtopten}$ in the tens column becomes a $\\var{10+newtopten}$. Now we can do&nbsp;$\\var{10+newtopten}-\\var{bot[1]}$, and write the result, $\\var{ansten}$, under the line in the tens column.</span>&nbsp;</p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td>\n<p><span data-jme-visible=\"tendiff&lt;0\">$\\overset{\\color{red}{\\var{newtophun}}}{\\var{top[2]}\\mkern-7.5mu\\color{red}{/}}$</span> <span data-jme-visible=\"tendiff&gt;=0\">$\\overset{\\phantom{1}}{\\var{top[2]}}$</span></p>\n</td>\n<td>\n<p><span data-jme-visible=\"unitdiff&lt;0 and tendiff&lt;0\">$\\overset{\\color{red}{1}\\var{newtopten}}{\\var{top[1]}\\mkern-7.5mu/}$</span> <span data-jme-visible=\"unitdiff&lt;0 and tendiff&gt;=0\">$\\overset{\\var{newtopten}}{\\var{top[1]}\\mkern-7.5mu/}$</span> <span data-jme-visible=\"unitdiff&gt;=0 and tendiff&lt;0\">$\\color{red}{^1}\\overset{\\phantom{1}}{\\var{top[1]}}$</span> <span data-jme-visible=\"unitdiff&gt;=0 and tendiff&gt;=0\">$\\overset{\\phantom{1}}{\\var{top[1]}}$</span></p>\n</td>\n<td><span data-jme-visible=\"unitdiff&lt;0\">${^1}\\overset{\\phantom{1}}{\\var{top[0]}}$</span> <span data-jme-visible=\"unitdiff&gt;=0\">$\\overset{\\phantom{1}}{\\var{top[0]}}$</span></td>\n<td>$-$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\var{bot[2]}$</td>\n<td>$\\var{bot[1]}$</td>\n<td>$\\var{bot[0]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td></td>\n<td>$\\color{red}{\\var{ansten}}$</td>\n<td>$\\var{ansunit}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>Now we&nbsp;try to subtract the digits in the hundreds column.</p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td>\n<p><span data-jme-visible=\"tendiff&lt;0\">$\\overset{\\var{newtophun}}{\\var{top[2]}\\mkern-7.5mu/}$</span> <span data-jme-visible=\"tendiff&gt;=0\">$\\overset{\\phantom{1}}{\\var{top[2]}}$</span></p>\n</td>\n<td>\n<p><span data-jme-visible=\"unitdiff&lt;0 and tendiff&lt;0\">$\\overset{{1}\\var{newtopten}}{\\var{top[1]}\\mkern-7.5mu/}$</span> <span data-jme-visible=\"unitdiff&lt;0 and tendiff&gt;=0\">$\\overset{\\var{newtopten}}{\\var{top[1]}\\mkern-7.5mu/}$</span> <span data-jme-visible=\"unitdiff&gt;=0 and tendiff&lt;0\">${^1}\\overset{\\phantom{1}}{\\var{top[1]}}$</span> <span data-jme-visible=\"unitdiff&gt;=0 and tendiff&gt;=0\">$\\overset{\\phantom{1}}{\\var{top[1]}}$</span></p>\n</td>\n<td><span data-jme-visible=\"unitdiff&lt;0\">${^1}\\overset{\\phantom{1}}{\\var{top[0]}}$</span> <span data-jme-visible=\"unitdiff&gt;=0\">$\\overset{\\phantom{1}}{\\var{top[0]}}$</span></td>\n<td>$-$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\var{bot[2]}$</td>\n<td>$\\var{bot[1]}$</td>\n<td>$\\var{bot[0]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td>$\\color{red}{\\var{anshun}}$</td>\n<td>$\\var{ansten}$</td>\n<td>$\\var{ansunit}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>The answer is therefore $\\var{ans}$.</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": "ans", "maxValue": "ans", "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": "Lengths of vectors and dot products", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Jessica Claridge", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1728/"}], "tags": [], "metadata": {"description": "<p>Calculations of the lengths of two 3D vectors, the distance between their terminal points, their sum, difference, and dot and cross products.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Given the vectors $\\boldsymbol{a}=\\pmatrix{\\var{a[0]},\\var{a[1]},\\var{a[2]}}$ and&nbsp;$\\boldsymbol{b}=\\pmatrix{\\var{b[0]},\\var{b[1]},\\var{b[2]}}$&nbsp;find:</p>", "advice": "<p>For the general 3-component vectors $\\boldsymbol{a}=\\pmatrix{a_1,a_2,a_3}$ and $\\boldsymbol{b}=\\pmatrix{b_1,b_2,b_3}$, we have</p>\n    <p>a)</p>\n    <p>Lengths: $a=\\lvert\\boldsymbol{a}\\rvert=\\sqrt{a_1^2+a_2^2+a_3^2}$ and $b=\\lvert\\boldsymbol{b}\\rvert=\\sqrt{b_1^2+b_2^2+b_3^2}$, which are <strong>scalar</strong> quantities.</p>\n    <p>&nbsp;</p>\n    <p>b)</p>\n    <p>Distance between the terminal points: $d=\\sqrt{(a_1-b_1)^2+(a_2-b_2)^2+(a_3-b_3)^2}$, which is a&nbsp;<strong>scalar</strong> quantity.</p>\n    <p>&nbsp;</p>\n    <p>c)</p>\n    <p>Sum $\\boldsymbol{a}+\\boldsymbol{b}=\\pmatrix{a_1+b_1,a_2+b_2,a_3+b_3}$ and difference $\\boldsymbol{a}-\\boldsymbol{b}=\\pmatrix{a_1-b_1,a_2-b_2,a_3-b_3}$, which are&nbsp;<strong>vector</strong> quantities.</p>\n    <p>&nbsp;</p>\n    <p>d)</p>\n    <p>Dot product: $\\boldsymbol{a\\cdot b}=a_1b_1+a_2b_2+a_3b_3$, which is a&nbsp;<strong>scalar</strong> quantity.</p>\n    <p>&nbsp;</p>\n    <p>e)</p>\n    <p>Cross product: $\\boldsymbol{a}\\times\\boldsymbol{b}=\\pmatrix{a_2b_3-a_3b_2,a_3b_1-a_1b_3,a_1b_2-a_2b_1}$, which is a&nbsp;<strong>vector</strong> quantity.</p>\n    <p>&nbsp;</p>\n    <p>In this question, therefore, we have:</p>\n    <p>a)</p>\n    <p>Lengths: $a=\\lvert\\boldsymbol{a}\\rvert=\\sqrt{\\var{a[0]^2}+\\var{a[1]^2}+\\var{a[2]^2}}=\\var{lena}$ and $b=\\lvert\\boldsymbol{b}\\rvert=\\sqrt{\\var{b[0]^2}+\\var{b[1]^2}+\\var{b[2]^2}}=\\var{lenb}$.</p>\n    <p>&nbsp;</p>\n    <p>b)</p>\n    <p>Distance between the terminal points: $d=\\sqrt{(\\simplify[std]{{a[0]}-{b[0]}})^2+(\\simplify[std]{{a[1]}-{b[1]}})^2+(\\simplify[std]{{a[2]}-{b[2]}})^2}=\\var{dist}$.</p>\n    <p>&nbsp;</p>\n    <p>c)</p>\n    <p>Sum $\\boldsymbol{a}+\\boldsymbol{b}=\\pmatrix{\\simplify[std]{{a[0]}+{b[0]}},\\simplify[std]{{a[1]}+{b[1]}},\\simplify[std]{{a[2]}+{b[2]}}}=\\pmatrix{\\var{sumab[0]},\\var{sumab[1]},\\var{sumab[2]}}$ and difference $\\boldsymbol{a}-\\boldsymbol{b}=\\pmatrix{\\simplify[std]{{a[0]}-{b[0]}},\\simplify[std]{{a[1]}-{b[1]}},\\simplify[std]{{a[2]}-{b[2]}}}=\\pmatrix{\\var{diffab[0]},\\var{diffab[1]},\\var{diffab[2]}}$.</p>\n    <p>&nbsp;</p>\n    <p>d)</p>\n    <p>Dot product: $\\boldsymbol{a\\cdot b}=(\\var{a[0]}\\times\\var{b[0]})+(\\var{a[1]}\\times\\var{b[1]})+(\\var{a[2]}\\times\\var{b[2]})=\\var{dotab}$.</p>\n    <p>&nbsp;</p>\n    <p>e)</p>\n    <p>Cross product: $\\boldsymbol{a}\\times\\boldsymbol{b}=\\pmatrix{\\simplify[std]{{a[1]*b[2]}-{a[2]*b[1]}},\\simplify[std]{{a[2]*b[0]}-{a[0]*b[2]}},\\simplify[std]{{a[0]*b[1]}-{a[1]*b[0]}}}=\\pmatrix{\\var{crossab[0]},\\var{crossab[1]},\\var{crossab[2]}}$.</p>", "rulesets": {"std": ["all", "!collectNumbers", "!noLeadingMinus"]}, "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "vector(repeat(random(1..9)*sign(random(1,-1)),3))", "description": "", "templateType": "anything"}, "crossab": {"name": "crossab", "group": "Ungrouped variables", "definition": "cross(a,b)", "description": "", "templateType": "anything"}, "lena": {"name": "lena", "group": "Ungrouped variables", "definition": "precround(abs(a),2)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "vector(repeat(random(1..9)*sign(random(1,-1)),3))", "description": "", "templateType": "anything"}, "dotab": {"name": "dotab", "group": "Ungrouped variables", "definition": "dot(a,b)", "description": "", "templateType": "anything"}, "sumab": {"name": "sumab", "group": "Ungrouped variables", "definition": "a+b", "description": "", "templateType": "anything"}, "dist": {"name": "dist", "group": "Ungrouped variables", "definition": "precround(abs(a-b),2)", "description": "", "templateType": "anything"}, "lenb": {"name": "lenb", "group": "Ungrouped variables", "definition": "precround(abs(b),2)", "description": "", "templateType": "anything"}, "diffab": {"name": "diffab", "group": "Ungrouped variables", "definition": "a-b", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "lenb", "lena", "b", "dist", "dotab", "diffab", "sumab", "crossab"], "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>Their lengths: $a=\\lvert\\boldsymbol{a}\\rvert=$ [[0]], $b=\\lvert\\boldsymbol{b}\\rvert=$ [[1]]. &nbsp;(Enter your answers to 2d.p.)</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": "lena-0.01", "maxValue": "lena+0.01", "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, "minValue": "lenb-0.01", "maxValue": "lenb+0.01", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "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>Their dot product $\\boldsymbol{a\\cdot b}=$ [[0]].</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": "dotab", "maxValue": "dotab", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Dot and cross product combinations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "parts": [{"displayType": "radiogroup", "layout": {"type": "all", "expression": ""}, "choices": ["$\\boldsymbol{(v\\cdot w)\\cdot u}$", "$\\boldsymbol{(v\\cdot w)u}$", "$\\boldsymbol{(v\\cdot w)\\times u}$", "$\\boldsymbol{(v\\times w)\\times u}$", "<p>$\\boldsymbol{(v\\times w)\\cdot u}$</p>"], "variableReplacementStrategy": "originalfirst", "matrix": [[0, 0, 0.4], [0, 0.4, 0], [0, 0, 0.4], [0, 0.4, 0], [0.4, 0, 0]], "shuffleChoices": true, "type": "m_n_x", "maxAnswers": 0, "marks": 0, "warningType": "none", "scripts": {}, "minMarks": 0, "minAnswers": 0, "maxMarks": 0, "shuffleAnswers": true, "showCorrectAnswer": true, "variableReplacements": [], "answers": ["<p>Scalar</p>", "<p>Vector</p>", "<p>Undefined</p>"]}], "variables": {}, "ungrouped_variables": [], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "functions": {}, "variable_groups": [], "showQuestionGroupNames": false, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Given the vectors $\\boldsymbol{v}$, $\\boldsymbol{w}$, $\\boldsymbol{u}$ in $\\mathbb{R}^3$, state whether the following quantities are scalars (real numbers), vectors (elements of $\\mathbb{R}^3$) or undefined.</p>\n<p>In this question, the&nbsp;symbol $\\cdot$ denotes the inner product and $\\times$ always denotes the cross product.</p>", "tags": ["checked2015", "cross product", "dot product", "inner product", "MAS1602", "mas1602", "scalar product", "scalars", "unused", "vector", "Vector", "vector product", "vectors"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n        \t\t<p><strong>15/07/2012:</strong></p>\n        \t\t<p>Added tags.</p>\n        \t\t<p><strong>16/07/2012:</strong></p>\n        \t\t<p><strong>Added tags.<br /></strong></p>\n        \t\t<p><strong>&nbsp;</strong></p>\n        \t\t<p><strong>&nbsp;</strong></p>\n        \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Determine if various combinations of vectors are defined or not.</p>"}, "advice": "<p>1. $\\boldsymbol{(v\\cdot w)\\cdot u}$ is <b>undefined</b> as $\\boldsymbol{v\\cdot w}$ is a scalar and we cannot take the inner product of a scalar with the vector $\\boldsymbol{u}$.</p>\n<p>2. $\\boldsymbol{(v\\cdot w) u}$ is a <b>vector</b> and is a scalar multiple of $\\boldsymbol{u}$ as $\\boldsymbol{v \\cdot w}$ is a scalar.</p>\n<p>3. $\\boldsymbol{(v \\cdot w)\\times u}$ is <b>undefined</b> as $\\boldsymbol{v\\cdot w}$ is a scalar and the cross product is only defined between vectors.</p>\n<p>4. $\\boldsymbol{(v\\times w)\\times u}$ is a <b>vector</b> as $\\boldsymbol{v \\times w}$ and $\\boldsymbol{u}$ are vectors and the cross product between vectors produces a vector.</p>\n<p>5. $\\boldsymbol{(v\\times w)\\cdot u}$ is a <b>scalar</b> as $\\boldsymbol{v \\times w}$ and $\\boldsymbol{u}$ are vectors and the inner or dot product is between vectors and produces a scalar.</p>"}, {"name": "Vectors: Scalar Product 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}], "tags": [], "metadata": {"description": "<p class=\"p1\">Given three 2-dimensional vectors $\\mathbf a$, $\\mathbf b$ and $\\mathbf c$, calculate the scalar product between $\\mathbf a$ and $\\mathbf b$, the angle between $\\mathbf a$ and $\\mathbf b$, and $\\mathbf a (\\mathbf b \\cdot \\mathbf c)$,</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Given the vectors</p>\n<p>\\[ \\mathbf a = \\var{a},\\quad \\mathbf b&nbsp; = \\var{b}, \\quad \\mathbf c = \\var{c}; \\]</p>\n<p></p>", "advice": "<p>To answer these questions, we want to use the equations for the scalar product. Recall:</p>\n<p>For the vectors $ \\mathbf v = \\pmatrix{v_1 \\\\ v_2},\\, \\mathbf w = \\pmatrix{w_1 \\\\ w_2},$</p>\n<p>\\[ \\begin{split} \\mathbf{v \\cdot w} &amp;\\,= v_1 \\times w_1 + v_2 \\times w_2 \\\\\\\\ \\mathbf{v \\cdot w} &amp;\\,= |\\mathbf v| |\\mathbf w | \\cos(\\theta), \\end{split} \\]</p>\n<p>where $|\\mathbf v|$ and $|\\mathbf w|$ are the magnitudes of the vectors, and $\\theta$ is the angle between the vectors.</p>\n<p>So, for the vectors $\\mathbf a = \\var{a}$ and $ \\mathbf b&nbsp; = \\var{b}$,</p>\n<p>\\[ \\begin{split} |\\mathbf a| &amp;\\,= \\simplify[!collectNumbers]{sqrt({a[0]}^2+{a[1]}^2)}&nbsp; \\\\ &amp;\\,=\\var{precround(length(a),2)}, \\end{split} \\]</p>\n<p>\\[ \\begin{split} |\\mathbf b| &amp;\\,= \\simplify[!collectNumbers]{sqrt({b[0]}^2+{b[1]}^2)}&nbsp; \\\\ &amp;\\,=\\var{precround(length(b),2)}, \\end{split} \\]</p>\n<p>and</p>\n<p>\\[ \\begin{split} \\mathbf{a \\cdot b} &amp;\\,= \\simplify[alwaysTimes,!collectNumbers]{{a[0]}*{b[0]}+{a[1]}*{b[1]}} \\\\ &amp;\\,= \\var{adotb}. \\end{split} \\]</p>\n<p>Using these results to find the angle between&nbsp;$\\mathbf a$ and&nbsp;$\\mathbf b$:</p>\n<p>\\[ \\begin{split} \\cos(\\theta) &amp;\\,= \\frac{\\mathbf{a \\cdot b}}{|\\mathbf a| |\\mathbf b |} , \\\\ \\\\ &amp;\\,=\\frac{\\var{adotb}}{\\var{precround(length(a),2)}\\times\\var{precround(length(b),2)}} \\\\\\\\ &amp;\\,=\\var{precround(adotb/(length(a)*length(b)),2)}.\\end{split} \\]</p>\n<p>Therefore, \\[ \\begin{split} \\theta &amp;\\,= \\cos^{-1}(\\var{precround(adotb/(length(a)*length(b)),2)}) \\\\ &amp;\\,= \\var{angleab}^\\circ . \\end{split}&nbsp;\\]</p>\n<p>To calculate $\\mathbf a \\left( \\mathbf{b \\cdot c} \\right)$, we want to first find the dot product between $\\mathbf b$ and $\\mathbf c$:</p>\n<p>\\[ \\begin{split} \\mathbf{b \\cdot c} &amp;\\,= \\simplify[alwaysTimes,!collectNumbers]{{b[0]}*{c[0]}+{b[1]}*{c[1]}} \\\\ &amp;\\,= \\var{dot(b,c)}. \\end{split} \\]</p>\n<p>Therefore,</p>\n<p>\\[ \\begin{split}&nbsp;\\mathbf a \\left( \\mathbf{b \\cdot c} \\right) &amp;\\,=&nbsp;&nbsp;\\var{dot(b,c)} \\var{a} \\\\\\\\ &amp;\\,= \\var{abdotc}. \\end{split} \\]</p>\n<p></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "vector(random(1..7),random(-4..4 except [0,1]))", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "vector(random(-2..7),random(3..8))", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "vector(random(-4..3 except [a[0],b[0]]),random(2..9))", "description": "", "templateType": "anything", "can_override": false}, "angleab": {"name": "angleab", "group": "Ungrouped variables", "definition": 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"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 angle between vectors $\\mathbf a$ and $\\mathbf b$:</p>\n<p>$\\theta = $[[0]]$^\\circ$</p>\n<p>(<em>Give your answer to 2 decimal places where necessary)</em></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": "{angleab}", "showPreview": true, 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