// Numbas version: finer_feedback_settings
{"name": "Vectors: Unit Vectors 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Vectors: Unit Vectors 1", "tags": [], "metadata": {"description": "<p>Find the unit vectors in the direction of four 2-dimensional vectors. Three of the vectors are given, and the fourth is expressed as a linear combination of two of the other vectors.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Find the unit vectors in the direction of</p>\n<p>\\[ \\mathbf a = \\var{a}, \\quad \\mathbf b = \\var{b}, \\quad \\mathbf c = \\var{c}, \\quad \\mathbf d = \\simplify{{m}*mathbf:a +{n}*mathbf:b}.\\]</p>", "advice": "<p>A unit vector in the direction of a vector $\\mathbf n$ is defined as&nbsp;</p>\n<p>\\[ \\mathbf{\\hat{n}} = \\dfrac{\\mathbf{n}}{|\\mathbf n |} , \\]</p>\n<p>where $|\\mathbf n |$ is the magnitude of $\\mathbf n$.</p>\n<p>For the vector $\\mathbf a = \\var{a}$,&nbsp;</p>\n<p>\\[ \\begin{split} | \\mathbf a | &amp;\\, = \\simplify[!collectNumbers]{sqrt({a[0]}^2+{a[1]}^2)} \\\\ &amp;\\, = \\simplify{sqrt({length(a)^2})}&nbsp;.\\end{split} \\]</p>\n<p>Therefore,</p>\n<p>\\[ \\mathbf{\\hat{a}} = \\frac{1}{\\simplify{sqrt({length(a)^2})}}\\var{a} = \\var{unita}. \\]</p>\n<p>Similarly,</p>\n<p>\\[ \\mathbf{\\hat{b}} = \\frac{1}{\\simplify{sqrt({length(b)^2})}}\\var{b} = \\var{unitb}, \\]</p>\n<p>and</p>\n<p>\\[ \\mathbf{\\hat{c}} = \\frac{1}{\\simplify{sqrt({length(c)^2})}}\\var{c} = \\var{unitc}. \\]</p>\n<p>For the vector $\\mathbf d = \\simplify{{m}*mathbf:a +{n}*mathbf:b}$:</p>\n<p>\\[ \\mathbf d = \\simplify[!collectNumbers]{{m}*{a} +{n}*{b}} = \\var{d}. \\]</p>\n<p>Therefore,</p>\n<p>\\[ \\mathbf{\\hat{d}} = \\frac{1}{\\simplify{sqrt({length(d)^2})}}\\var{d} = \\var{unitd}. \\]</p>\n<p></p>\n<p>\\[\\]</p>\n<p>Note: It is also acceptable to&nbsp;give the answer in the form \\[ \\mathbf{\\hat{n}} = \\dfrac{1}{|\\mathbf n |}\\mathbf{n}, \\]</p>\n<p>rather than dividing each term by the magnitude.</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "vector(repeat(random(-5..5 except 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