// Numbas version: finer_feedback_settings
{"name": "Vector equation of a line", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"w": {"group": "Ungrouped variables", "templateType": "anything", "definition": "matrix([c,d,f])", "description": "", "name": "w"}, "s1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s1"}, "v": {"group": "Ungrouped variables", "templateType": "anything", "definition": "matrix([a,b,g])", "description": "", "name": "v"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(2..9)", "description": "", "name": "a"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s3*random(2..9)", "description": "", "name": "c"}, "s4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s4"}, "s2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s2"}, "a1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a+lam*c-mu*al", "description": "", "name": "a1"}, "mu": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s2*random(1..5)", "description": "", "name": "mu"}, "be": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-5..5)", "description": "", "name": "be"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s2*random(2..9)", "description": "", "name": "b"}, "p": {"group": "Ungrouped variables", "templateType": "anything", "definition": "vector(a,b,g)+lam*vector(c,d,f)", "description": "<p>Point of intersection of the two lines</p>", "name": "p"}, "f1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "ga", "description": "", "name": "f1"}, "g1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "g+lam*f-mu*ga", "description": "", "name": "g1"}, "lam": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s3*random(1..5)", "description": "", "name": "lam"}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s4*random(2..9)", "description": "", "name": "d"}, "f": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "description": "", "name": "f"}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "b+lam*d-mu*be", "description": "", "name": "b1"}, "al": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-5..5)", "description": "", "name": "al"}, "d1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "be", "description": "", "name": "d1"}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "al", "description": "", "name": "c1"}, "ga": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-5..5)", "description": "", "name": "ga"}, "s3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s3"}, "g": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(2..9)", "description": "", "name": "g"}}, "ungrouped_variables": ["a", "a1", "al", "b", "b1", "be", "c", "c1", "d", "d1", "f", "f1", "g", "g1", "ga", "lam", "mu", "s1", "s2", "s3", "s4", "v", "w", "p"], "preamble": {"css": "", "js": ""}, "name": "Vector equation of a line", "variable_groups": [], "functions": {}, "parts": [{"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "<p>Find the vector equation of Line 1, which passes through the points $\\boldsymbol{x_0}$ and&nbsp;$\\boldsymbol{x_1}$.</p>\n<p>Input the vector equation in the form:</p>\n<p>\\[\\boldsymbol{r} = \\begin{pmatrix} a_1 \\\\ a_2 \\\\ a_3 \\end{pmatrix}&nbsp;+ \\lambda \\begin{pmatrix} b_1 \\\\ b_2 \\\\ b_3 \\end{pmatrix} \\]</p>\n<p>such that $\\boldsymbol{r} = \\boldsymbol{x_0}$ when $\\lambda=0$ and $\\boldsymbol{r}=\\boldsymbol{x_1}$ when $\\lambda=1$ by filling in the appropriate fields below:</p>\n<p>$ \\boldsymbol{r} = $&nbsp;[[0]]&nbsp;$ + \\lambda $&nbsp;[[1]]</p>", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "correctAnswer": "vector(a,b,g)", "allowResize": false, "unitTests": [], "correctAnswerFractions": false, "variableReplacementStrategy": "originalfirst", "numRows": "3", "scripts": {}, "type": "matrix", "numColumns": 1, "tolerance": 0, "markPerCell": false, "variableReplacements": [], "marks": "0.75", "showFeedbackIcon": true}, {"showCorrectAnswer": true, "allowFractions": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "correctAnswer": "vector(c,d,f)", "allowResize": false, "unitTests": [], "correctAnswerFractions": false, "variableReplacementStrategy": "originalfirst", "numRows": "3", "scripts": {}, "type": "matrix", "numColumns": 1, "tolerance": 0, "markPerCell": false, "variableReplacements": [], "marks": "0.75", "showFeedbackIcon": true}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}, {"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "<p>Now find the vector equation of Line 2, which passes through the points $\\boldsymbol{y_0}$ and $\\boldsymbol{y_1}$ in the form</p>\n<p>\\[ \\boldsymbol{r} = \\begin{pmatrix} c_1 \\\\ c_2 \\\\ c_3 \\end{pmatrix} + \\mu \\begin{pmatrix} d_1 \\\\ d_2 \\\\ d_3 \\end{pmatrix} \\]</p>\n<p>such that $\\boldsymbol{r}=\\boldsymbol{y_0}$ when $\\mu=0$ and $\\boldsymbol{r}=\\boldsymbol{y_1}$ when $\\mu=1$ by filling in the appropriate fields below:</p>\n<p>$ \\boldsymbol{r} = $&nbsp;[[0]] $ + \\mu $&nbsp;[[1]]</p>", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "correctAnswer": "vector(a1,b1,g1)", "allowResize": false, "unitTests": [], "correctAnswerFractions": false, "variableReplacementStrategy": "originalfirst", "numRows": "3", "scripts": {}, "type": "matrix", "numColumns": 1, "tolerance": 0, "markPerCell": false, "variableReplacements": [], "marks": "0.75", "showFeedbackIcon": true}, {"showCorrectAnswer": true, "allowFractions": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "correctAnswer": "vector(c1,d1,f1)", "allowResize": false, "unitTests": [], "correctAnswerFractions": false, "variableReplacementStrategy": "originalfirst", "numRows": "3", "scripts": {}, "type": "matrix", "numColumns": 1, "tolerance": 0, "markPerCell": false, "variableReplacements": [], "marks": "0.75", "showFeedbackIcon": true}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}, {"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "<p>You are told&nbsp;that Line 1 and Line 2 intersect in a point $\\boldsymbol{P}$.</p>\n<p>Find $\\boldsymbol{P}$.</p>\n<p>$\\boldsymbol{P} = $&nbsp;[[0]]</p>", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "correctAnswer": "p", "allowResize": false, "unitTests": [], "correctAnswerFractions": false, "variableReplacementStrategy": "originalfirst", "numRows": "3", "scripts": {}, "type": "matrix", "numColumns": 1, "tolerance": 0, "markPerCell": false, "variableReplacements": [], "marks": "3", "showFeedbackIcon": true}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}], "statement": "<p>You are given the vectors</p>\n<p>\\begin{align}<br/>\\boldsymbol{x_0} &amp;= \\var{vector(a,b,g)} , &amp; \\boldsymbol{x_1} &amp; = \\var{vector(a+c,b+d,g+f)}, \\\\[1em]<br/>\\boldsymbol{y_0} &amp;= \\var{vector(a1,b1,g1)}, &amp; \\boldsymbol{y_1} &amp;=\\var{vector(a1+c1,b1+d1,g1+f1)}<br/>\\end{align}</p>\n<p>in $\\mathbb{R^3}$.</p>", "tags": ["checked2015", "equation of a line", "equation of a line through a vector in the direction of another vector", "Finding a common point for two lines in three dimensional space", "intersection of two lines in three dimensional space", "lines in three dimensional space", "three dimensional geometry", "vector equation of a line", "vectors"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "extensions": [], "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given a pair of 3D position vectors,&nbsp;find the vector equation of the line through both.&nbsp; Find two such lines and their point of intersection.</p>"}, "advice": "<h4>a)</h4>\n<p>For $\\lambda=0$ we have \\[\\boldsymbol{r} = \\begin{pmatrix} a_1 \\\\ a_2 \\\\ a_3 \\end{pmatrix}\\]</p>\n<p>and we want this to be equal to $\\boldsymbol{x_0}$.&nbsp; So we need $a_1 = \\var{a}$, $a_2 = \\var{b}$, and $a_3 = \\var{g}$.</p>\n<p>For $\\lambda=1$ we need $\\boldsymbol{r}=\\boldsymbol{x_1}$, and so</p>\n<p>\\[\\begin{pmatrix} a_1 \\\\ a_2 \\\\ a_3 \\end{pmatrix} + \\begin{pmatrix} b_1 \\\\ b_2 \\\\ b_3 \\end{pmatrix} = \\var{vector(a+c,b+d,g+f)}\\]</p>\n<p>which tells us that $b_1=\\var{c}$, $b_2=\\var{d}$, and $b_3=\\var{f}$.&nbsp; Thus the equation for Line 1 is</p>\n<p>\\[\\boldsymbol{r} = \\var{vector(a,b,g)} + \\lambda \\var{vector(c,d,f)}\\]</p>\n<h4>b)</h4>\n<p>Proceeding as in part a), we find that</p>\n<p>\\[\\boldsymbol{r} = \\var{vector(a1,b1,g1)} + \\mu \\var{vector(c1,d1,f1)}\\]</p>\n<h4>c)</h4>\n<p>Write out a set of simultaneous equations for each component of $\\boldsymbol{P}$:</p>\n<p>\\begin{align}<br/>\\simplify[]{{a} + lambda*{c}} &amp;= \\simplify[]{{a1} + mu*{c1}} \\\\<br/>\\simplify[]{{b} + lambda*{d}} &amp;= \\simplify[]{{b1} + mu*{d1}} \\\\<br/>\\simplify[]{{g} + lambda*{f}} &amp;= \\simplify[]{{g1} + mu*{f1}}<br/>\\end{align}</p>\n<p>By solving these equations, we find that the point $\\boldsymbol{P}$ common to both lines is given by $\\lambda=\\var{lam},\\mu=\\var{mu}$, and</p>\n<p>\\[\\boldsymbol{P} = \\var{p}\\]</p>", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}