Point of intersection of the two lines
", "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": "Find the vector equation of Line 1, which passes through the points $\\boldsymbol{x_0}$ and $\\boldsymbol{x_1}$.
\nInput the vector equation in the form:
\n\\[\\boldsymbol{r} = \\begin{pmatrix} a_1 \\\\ a_2 \\\\ a_3 \\end{pmatrix} + \\lambda \\begin{pmatrix} b_1 \\\\ b_2 \\\\ b_3 \\end{pmatrix} \\]
\nsuch 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:
\n$ \\boldsymbol{r} = $ [[0]] $ + \\lambda $ [[1]]
", "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": "Now find the vector equation of Line 2, which passes through the points $\\boldsymbol{y_0}$ and $\\boldsymbol{y_1}$ in the form
\n\\[ \\boldsymbol{r} = \\begin{pmatrix} c_1 \\\\ c_2 \\\\ c_3 \\end{pmatrix} + \\mu \\begin{pmatrix} d_1 \\\\ d_2 \\\\ d_3 \\end{pmatrix} \\]
\nsuch 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:
\n$ \\boldsymbol{r} = $ [[0]] $ + \\mu $ [[1]]
", "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": "You are told that Line 1 and Line 2 intersect in a point $\\boldsymbol{P}$.
\nFind $\\boldsymbol{P}$.
\n$\\boldsymbol{P} = $ [[0]]
", "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": "You are given the vectors
\n\\begin{align}
\\boldsymbol{x_0} &= \\var{vector(a,b,g)} , & \\boldsymbol{x_1} & = \\var{vector(a+c,b+d,g+f)}, \\\\[1em]
\\boldsymbol{y_0} &= \\var{vector(a1,b1,g1)}, & \\boldsymbol{y_1} &=\\var{vector(a1+c1,b1+d1,g1+f1)}
\\end{align}
in $\\mathbb{R^3}$.
", "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": "Given a pair of 3D position vectors, find the vector equation of the line through both. Find two such lines and their point of intersection.
"}, "advice": "For $\\lambda=0$ we have \\[\\boldsymbol{r} = \\begin{pmatrix} a_1 \\\\ a_2 \\\\ a_3 \\end{pmatrix}\\]
\nand we want this to be equal to $\\boldsymbol{x_0}$. So we need $a_1 = \\var{a}$, $a_2 = \\var{b}$, and $a_3 = \\var{g}$.
\nFor $\\lambda=1$ we need $\\boldsymbol{r}=\\boldsymbol{x_1}$, and so
\n\\[\\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)}\\]
\nwhich tells us that $b_1=\\var{c}$, $b_2=\\var{d}$, and $b_3=\\var{f}$. Thus the equation for Line 1 is
\n\\[\\boldsymbol{r} = \\var{vector(a,b,g)} + \\lambda \\var{vector(c,d,f)}\\]
\nProceeding as in part a), we find that
\n\\[\\boldsymbol{r} = \\var{vector(a1,b1,g1)} + \\mu \\var{vector(c1,d1,f1)}\\]
\nWrite out a set of simultaneous equations for each component of $\\boldsymbol{P}$:
\n\\begin{align}
\\simplify[]{{a} + lambda*{c}} &= \\simplify[]{{a1} + mu*{c1}} \\\\
\\simplify[]{{b} + lambda*{d}} &= \\simplify[]{{b1} + mu*{d1}} \\\\
\\simplify[]{{g} + lambda*{f}} &= \\simplify[]{{g1} + mu*{f1}}
\\end{align}
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
\n\\[\\boldsymbol{P} = \\var{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/"}]}