// Numbas version: exam_results_page_options
{"name": "Vectors: when perpendicular 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "s3", "s2", "s1", "s5", "s4"], "name": "Vectors: when perpendicular 1", "tags": ["dot product", "finding perpendicular vectors", "inner product", "perpendicular vectors", "product", "scalar product", "vectors"], "preamble": {"css": "", "js": ""}, "advice": "<h4>a)</h4>\n<p>$\\boldsymbol{A}$ and $\\boldsymbol{B}$ are perpendicular to one another when $\\boldsymbol{A \\cdot B} = 0$.</p>\n<p>Now \\[ \\begin{eqnarray*}\\boldsymbol{A \\cdot B} &amp;=&amp; \\simplify[]{{a}*{c}+{b}*{d}+lambda*{f}}\\\\ &amp;=&amp; \\simplify[std]{{f}*lambda+{a*c+b*d}} \\end{eqnarray*} \\]<br/>Hence \\[\\boldsymbol{A \\cdot B} = 0 \\Rightarrow \\simplify[std]{{f}*lambda+{a*c+b*d}=0} \\Rightarrow \\lambda = \\simplify[std]{{-a*c-b*d}/{f}}\\]</p>\n<h4>b)</h4>\n<p>$\\boldsymbol{A}$ is in the $xy$ plane when $\\lambda=0$.</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "<p>Find $\\lambda$ such that $\\boldsymbol{A}$ and $\\boldsymbol{B}$ are perpendicular to one another:</p>\n<p>$\\lambda =\\;\\;$ [[0]].</p>\n<p>Enter your answer as a fraction or an integer and not as a decimal.</p>", "marks": 0, "gaps": [{"notallowed": {"message": "<p>Enter as a fraction or an integer and not as a decimal.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1.5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{-a*c-b*d}/{f}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>Find $\\lambda$ such that $\\boldsymbol{A}$ is in the $xy$ plane:</p>\n<p>$\\lambda =\\;\\;$ [[0]].</p>", "marks": 0, "gaps": [{"allowFractions": false, "marks": 0.5, "maxValue": "{0}", "minValue": "{0}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "<p>Given the vectors:<br/>\\[\\boldsymbol{A}=\\simplify[std]{{a}v:e_1+{b}v:e_2+lambda*v:e_3},\\;\\;\\;\\boldsymbol{B}=\\simplify[std]{{c}v:e_1+{d}v:e_2+{f}v:e_3}\\]</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "s1*random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "s3*random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "s2*random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "s4*random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "g": {"definition": "s1*random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "g", "description": ""}, "f": {"definition": "random(2,4,5,10)", "templateType": "anything", "group": "Ungrouped variables", "name": "f", "description": ""}, "s3": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s3", "description": ""}, "s2": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s2", "description": ""}, "s1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "s5": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s5", "description": ""}, "s4": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s4", "description": ""}}, "metadata": {"notes": "\n        \t\t<p><strong>15/07/2012:</strong></p>\n        \t\t<p>Added tags.</p>\n        \t\t<p>Last part is too easy.</p>\n        \t\t<p><strong>16/07/2012:</strong></p>\n        \t\t<p><strong>Added tags.</strong></p>\n        \t\t<p><strong>Question appears to be working correctly.<br /></strong></p>\n        \t\t<p><strong>Agree that last part is too easy.<br /></strong></p>\n        \t\t", "description": "<p><span class=\"Apple-style-span\" style=\"-webkit-tap-highlight-color: rgba(26, 26, 26, 0.296875); -webkit-composition-fill-color: rgba(175, 192, 227, 0.230469); -webkit-composition-frame-color: rgba(77, 128, 180, 0.230469);\"><span class=\"Apple-style-span\" style=\"-webkit-tap-highlight-color: rgba(26, 26, 26, 0.296875); -webkit-composition-fill-color: rgba(175, 192, 227, 0.230469); -webkit-composition-frame-color: rgba(77, 128, 180, 0.230469);\">When are vectors $\\boldsymbol{A,\\;B}$&nbsp;</span></span><span class=\"Apple-style-span\" style=\"-webkit-tap-highlight-color: rgba(26, 26, 26, 0.296875); -webkit-composition-fill-color: rgba(175, 192, 227, 0.230469); -webkit-composition-frame-color: rgba(77, 128, 180, 0.230469);\">perpendicular?</span></p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}