Top of the formula for the minimum distance: $(p-x_0) \\times v$.
", "name": "top"}, "f": {"group": "Vector v", "templateType": "anything", "definition": "random(2..9)", "description": "", "name": "f"}, "directions": {"group": "Vector p", "templateType": "anything", "definition": "map(id(3)[x],x,shuffle(0..2))", "description": "Shuffled list of axis vectors
", "name": "directions"}, "b": {"group": "Vector x_0", "templateType": "anything", "definition": "random(-1,1)*random(2..9)", "description": "", "name": "b"}, "a": {"group": "Vector x_0", "templateType": "anything", "definition": "s1*random(2..9)", "description": "", "name": "a"}, "g": {"group": "Vector x_0", "templateType": "anything", "definition": "s1*random(2..9)", "description": "", "name": "g"}, "p": {"group": "Vector p", "templateType": "anything", "definition": "random(-1,1)*directions[0] + random(-5..5 except -1..1)*directions[1]", "description": "", "name": "p"}, "x_0": {"group": "Vector x_0", "templateType": "anything", "definition": "vector(random(2..9),0,random(2..9))*random(-1,1) + vector(0,random(2..9),0)*random(-1,1)", "description": "", "name": "x_0"}, "c": {"group": "Vector v", "templateType": "anything", "definition": "random(-1,1)*random(2..9)", "description": "", "name": "c"}}, "ungrouped_variables": [], "name": "Minimum distance between a point and a line in 3D", "functions": {}, "preamble": {"css": "", "js": ""}, "parts": [{"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "steps": [{"variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "scripts": {}, "customMarkingAlgorithm": "", "type": "information", "showCorrectAnswer": true, "unitTests": [], "prompt": "The minimum distance between the line and the point is given by
\n\\[ \\frac{\\left\\lVert(\\boldsymbol{p} - \\boldsymbol{x_0})\\times \\boldsymbol{v} \\right\\rVert}{\\lVert \\boldsymbol{v} \\rVert}\\]
", "variableReplacements": [], "marks": 0}], "prompt": "Distance = [[0]]
\nEnter your answer exactly, using the function sqrt(x) if necessary. Do not use decimals.
You can get help by clicking on Show steps.
", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "sqrt({abs(top)^2}/{abs(v)^2})", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "showPreview": true, "notallowed": {"message": "Enter all numbers as integers, do not use decimals
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "unitTests": [], "checkVariableNames": false, "vsetRange": [0, 1], "vsetRangePoints": 5, "failureRate": 1, "scripts": {}, "answerSimplification": "std", "type": "jme", "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 2, "showFeedbackIcon": true}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "stepsPenalty": 0, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "In $\\mathbb{R}^3$ find the distance between the point $\\boldsymbol{p} = \\var{p}$ and the line through the point $\\boldsymbol{x_0} = \\var{x_0}$ that is parallel to the vector $\\boldsymbol{v} = \\var{v}$.
", "tags": ["checked2015", "cross product of vectors", "distance between a point and a line", "distance between two points", "equation of a line through a point and in the direction of a vector", "minimum distance", "minimum distance between a point and a line in three space", "modulus of a vector", "three dimensional vector geometry", "vector", "Vector", "vector geometry", "vector product", "vectors"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "extensions": [], "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Find minimum distance between a point and a line in 3-space. The line goes through a given point in the direction of a given vector.
\nThe correct solution is given, however the accuracy of 0.001 is not enough as in some cases answers near to the correct solution are also marked as correct.
"}, "advice": "The line through $\\boldsymbol{x_0} = \\var{x_0}$ in the direction of $\\boldsymbol{v}=\\var{v}$ has equation:
\n\\[ \\boldsymbol{r} = \\boldsymbol{x_0} + \\lambda \\boldsymbol{v} = \\var{x_0} + \\lambda \\var{v} \\]
\nThe minimum distance between this line and the point $\\boldsymbol{p} = \\var{p}$ is given by
\n\\[ \\frac{\\left\\lVert(\\boldsymbol{p}-\\boldsymbol{x_0}) \\times \\boldsymbol{v} \\right\\rVert}{\\lVert \\boldsymbol{v} \\rVert} \\]
\nNow,
\n\\begin{align}
\\boldsymbol{p} - \\boldsymbol{x_0} &= \\var{p-x_0} \\Rightarrow \\\\[1em]
(\\boldsymbol{p}-\\boldsymbol{x_0}) \\times \\boldsymbol{v} &= \\var{p-x_0} \\times \\var{v} \\\\[1em]
&= \\var{top}
\\end{align}
Since
\n\\begin{align}
\\left\\lVert \\begin{matrix} \\var{top[0]} \\\\ \\var{top[1]} \\\\ \\var{top[2]} \\end{matrix} \\right \\rVert &= \\simplify[]{sqrt({top[0]}^2 + {top[1]}^2 + {top[2]}^2)} \\\\ &= \\sqrt{\\var{abs(top)^2}}
\\end{align}
and $\\lVert \\boldsymbol{v} \\rVert = \\simplify[]{sqrt({v[0]}^2 + {v[1]}^2 + {v[2]}^2)} = \\sqrt{\\var{abs(v)^2}}$, the distance is then:
\n\\[\\simplify{sqrt({abs(top)^2}/{abs(v)^2})}\\]
", "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/"}]}