Elementary operations on vectors; sum, modulus, unit vector, scalar multiple.
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", "scripts": {}, "unitTests": [], "gaps": [{"tolerance": 0, "type": "matrix", "scripts": {}, "unitTests": [], "numColumns": 1, "extendBaseMarkingAlgorithm": true, "correctAnswer": "v", "showFeedbackIcon": true, "variableReplacements": [], "numRows": "3", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "correctAnswerFractions": false, "allowResize": false, "marks": "1", "customMarkingAlgorithm": "", "allowFractions": false, "markPerCell": false}], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "showCorrectAnswer": true, "marks": 0, "customMarkingAlgorithm": ""}, {"type": "gapfill", "prompt": "Calculate $\\boldsymbol{v}+3\\boldsymbol{w} = $[[0]]
", "scripts": {}, "unitTests": [], "gaps": [{"tolerance": 0, "type": "matrix", "scripts": {}, "unitTests": [], "numColumns": 1, "extendBaseMarkingAlgorithm": true, "correctAnswer": "v1 + 3*v2", "showFeedbackIcon": true, "variableReplacements": [], "numRows": "3", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "correctAnswerFractions": false, "allowResize": false, "marks": 1, "customMarkingAlgorithm": "", "allowFractions": false, "markPerCell": false}], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "showCorrectAnswer": true, "marks": 0, "customMarkingAlgorithm": ""}, {"type": "gapfill", "prompt": "Calculate $4\\boldsymbol{v}-2\\boldsymbol{w} = $[[0]]
", "scripts": {}, "unitTests": [], "gaps": [{"tolerance": 0, "type": "matrix", "scripts": {}, "unitTests": [], "numColumns": 1, "extendBaseMarkingAlgorithm": true, "correctAnswer": "4*v1 - 2*v2", "showFeedbackIcon": true, "variableReplacements": [], "numRows": "3", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "correctAnswerFractions": false, "allowResize": false, "marks": 1, "customMarkingAlgorithm": "", "allowFractions": false, "markPerCell": false}], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "showCorrectAnswer": true, "marks": 0, "customMarkingAlgorithm": ""}, {"type": "gapfill", "prompt": "Calculate the following.
\n$\\lVert \\boldsymbol{v} \\rVert=$ [[0]]
\n$\\lVert \\boldsymbol{w} \\rVert = $ [[1]]
\n$\\lVert \\boldsymbol{v}+\\boldsymbol{w} \\rVert = $ [[2]]
", "scripts": {}, "unitTests": [], "gaps": [{"vsetRange": [0, 1], "type": "jme", "expectedVariableNames": [], "checkingType": "absdiff", "answerSimplification": "std", "answer": "sqrt({a^2+b^2+g^2})", "unitTests": [], "showPreview": true, "extendBaseMarkingAlgorithm": true, "vsetRangePoints": 5, "showFeedbackIcon": true, "checkingAccuracy": 0.001, "variableReplacements": [], "failureRate": 1, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "checkVariableNames": false}, {"vsetRange": [0, 1], "type": "jme", "expectedVariableNames": [], "checkingType": "absdiff", "answerSimplification": "std", "answer": "sqrt({c^2+d^2+f^2})", "unitTests": [], "showPreview": true, "extendBaseMarkingAlgorithm": true, "vsetRangePoints": 5, "showFeedbackIcon": true, "checkingAccuracy": 0.001, "variableReplacements": [], "failureRate": 1, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "checkVariableNames": false}, {"vsetRange": [0, 1], "type": "jme", "expectedVariableNames": [], "checkingType": "absdiff", "answerSimplification": "std", "answer": "sqrt({(a+c)^2+ (b+d)^2 +(g+f)^2})", "unitTests": [], "showPreview": true, "extendBaseMarkingAlgorithm": true, "vsetRangePoints": 5, "showFeedbackIcon": true, "checkingAccuracy": 0.001, "variableReplacements": [], "failureRate": 1, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "checkVariableNames": false}], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "showCorrectAnswer": true, "marks": 0, "customMarkingAlgorithm": ""}, {"type": "gapfill", "prompt": "Let $\\boldsymbol{z}=\\boldsymbol{v}+\\boldsymbol{w}$.
\nCalculate the unit vector $\\boldsymbol{u_z}$ in the direction of $\\boldsymbol{z}$. Write $\\boldsymbol{u_z}$ as a row vector.
\n$\\boldsymbol{u_z}= \\big($ [[0]], [[1]], [[2]] $\\big)$
\nYou must enter your answers exactly, using the function sqrt(x) if necessary.
Calculate
\n$\\var{a4}\\boldsymbol{v} = $ [[0]]
\n$\\var{b4}\\boldsymbol{w} = $ [[1]]
", "scripts": {}, "unitTests": [], "gaps": [{"tolerance": 0, "type": "matrix", "scripts": {}, "unitTests": [], "numColumns": 1, "extendBaseMarkingAlgorithm": true, "correctAnswer": "a4*v1", "showFeedbackIcon": true, "variableReplacements": [], "numRows": "3", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "correctAnswerFractions": false, "allowResize": false, "marks": "1", "customMarkingAlgorithm": "", "allowFractions": false, "markPerCell": false}, {"tolerance": 0, "type": "matrix", "scripts": {}, "unitTests": [], "numColumns": 1, "extendBaseMarkingAlgorithm": true, "correctAnswer": "b4*v2", "showFeedbackIcon": true, "variableReplacements": [], "numRows": "3", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "correctAnswerFractions": false, "allowResize": false, "marks": "1", "customMarkingAlgorithm": "", "allowFractions": false, "markPerCell": false}], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "showCorrectAnswer": true, "marks": 0, "customMarkingAlgorithm": ""}, {"type": "gapfill", "prompt": "Calculate the unit vector $\\boldsymbol{u_v}$ parallel to $\\boldsymbol{v}$, and the unit vector $-\\boldsymbol{u_w}$ anti-parallel to $\\boldsymbol{w}$. Write both vectors as row vectors.
\n$\\boldsymbol{u_v} = \\big($ [[0]], [[1]], [[2]] $\\big)$
\n$-\\boldsymbol{u_w} = \\big($ [[3]], [[4]], [[5]] $\\big)$
", "scripts": {}, "unitTests": [], "gaps": [{"vsetRange": [0, 1], "type": "jme", "expectedVariableNames": [], "checkingType": "absdiff", "answerSimplification": "std", "answer": "({a} / Sqrt({(({a} ^ 2) + ({b} ^ 2) + ({g} ^ 2))}))", "unitTests": [], "showPreview": true, "extendBaseMarkingAlgorithm": true, "vsetRangePoints": 5, "showFeedbackIcon": true, "checkingAccuracy": 0.001, "variableReplacements": [], "failureRate": 1, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "checkVariableNames": false}, {"vsetRange": [0, 1], "type": "jme", "expectedVariableNames": [], "checkingType": "absdiff", "answerSimplification": "std", "answer": "({b} / Sqrt({(({a} ^ 2) + ({b} ^ 2) + ({g} ^ 2))}))", "unitTests": [], "showPreview": true, "extendBaseMarkingAlgorithm": true, "vsetRangePoints": 5, "showFeedbackIcon": true, "checkingAccuracy": 0.001, "variableReplacements": [], "failureRate": 1, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "checkVariableNames": false}, {"vsetRange": [0, 1], "type": "jme", "expectedVariableNames": [], "checkingType": "absdiff", "answerSimplification": "std", "answer": "({g} / Sqrt({(({a} ^ 2) + ({b} ^ 2) + ({g} ^ 2))}))", "unitTests": [], "showPreview": true, "extendBaseMarkingAlgorithm": true, "vsetRangePoints": 5, "showFeedbackIcon": true, "checkingAccuracy": 0.001, "variableReplacements": [], "failureRate": 1, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "checkVariableNames": false}, {"vsetRange": [0, 1], "type": "jme", "expectedVariableNames": [], "checkingType": "absdiff", "answerSimplification": "std", "answer": "({( - c)} / Sqrt({(({c} ^ 2) + ({d} ^ 2) + ({f} ^ 2))}))", "unitTests": [], "showPreview": true, "extendBaseMarkingAlgorithm": true, "vsetRangePoints": 5, "showFeedbackIcon": true, "checkingAccuracy": 0.001, "variableReplacements": [], "failureRate": 1, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "checkVariableNames": false}, {"vsetRange": [0, 1], "type": "jme", "expectedVariableNames": [], "checkingType": "absdiff", "answerSimplification": "std", "answer": "({( - d)} / Sqrt({(({c} ^ 2) + ({d} ^ 2) + ({f} ^ 2))}))", "unitTests": [], "showPreview": true, "extendBaseMarkingAlgorithm": true, "vsetRangePoints": 5, "showFeedbackIcon": true, "checkingAccuracy": 0.001, "variableReplacements": [], "failureRate": 1, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "checkVariableNames": false}, {"vsetRange": [0, 1], "type": "jme", "expectedVariableNames": [], "checkingType": "absdiff", "answerSimplification": "std", "answer": "({( - f)} / Sqrt({(({c} ^ 2) + ({d} ^ 2) + ({f} ^ 2))}))", "unitTests": [], "showPreview": true, "extendBaseMarkingAlgorithm": true, "vsetRangePoints": 5, "showFeedbackIcon": true, "checkingAccuracy": 0.001, "variableReplacements": [], "failureRate": 1, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "checkVariableNames": false}], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "showCorrectAnswer": true, "marks": 0, "customMarkingAlgorithm": ""}], "advice": "\\[\\boldsymbol{v}+\\boldsymbol{w} = \\var{vector(a,b,g)} + \\var{vector(c,d,f)} = \\var{vector(a+c,b+d,g+f)} \\]
\nIn general for a vector $\\boldsymbol{x}= \\begin{pmatrix}x_1 \\\\ x_2 \\\\ x_3 \\end{pmatrix}$, we have $\\lVert \\boldsymbol{x} \\rVert = \\sqrt{x_1^2+x_2^2+x_3^2}$.
\nHence:
\n\\begin{align}
\\lVert \\boldsymbol{v} \\rVert &= \\sqrt{\\var{a^2}+\\var{b^2}+\\var{g^2}} = \\simplify[all]{ sqrt({a^2+b^2+g^2})} \\\\
\\lVert \\boldsymbol{w} \\rVert &= \\sqrt{\\var{c^2}+\\var{d^2}+\\var{f^2}} = \\simplify[all]{ sqrt({c^2+d^2+f^2})} \\\\
\\lVert \\boldsymbol{v+w} \\rVert &= \\sqrt{\\var{(a+c)^2}+\\var{(b+d)^2}+\\var{(g+f)^2}} = \\simplify[all]{ sqrt({(a+c)^2+(b+d)^2+(f+g)^2})}
\\end{align}
Given a vector $\\boldsymbol{x}= \\begin{pmatrix} x_1 \\\\ x_2 \\\\ x_3 \\end{pmatrix}$, the unit vector parallel to $\\boldsymbol{x}$ is given by:
\n\\[ \\boldsymbol{u_x} = \\frac{1}{\\lVert \\boldsymbol{x} \\rVert} \\begin{pmatrix} x_1 \\\\ x_2 \\\\ x_3 \\end{pmatrix} = \\begin{pmatrix} \\frac{x_1}{\\lVert \\boldsymbol{x} \\rVert} \\\\ \\frac{x_2}{\\lVert \\boldsymbol{x} \\rVert} \\\\ \\frac{x_3}{\\lVert \\boldsymbol{x} \\rVert} \\end{pmatrix} \\]
\nFor this example we have $\\lVert \\boldsymbol{v+w} \\rVert =\\simplify[std]{sqrt({(a+c)^2+(b+d)^2+(f+g)^2})}$, hence:
\n\\begin{align}
&&\\boldsymbol{z} = \\boldsymbol{v} + \\boldsymbol{w} &= \\begin{pmatrix} \\var{a+c} \\\\ \\var{b+d} \\\\ \\var{g+f} \\end{pmatrix} \\\\
\\implies && \\boldsymbol{u_z} &= \\frac{1}{\\sqrt{\\var{ssquares}}} \\begin{pmatrix} \\var{a+c} \\\\ \\var{b+d} \\\\ \\var{g+f} \\end{pmatrix} \\\\[1em]
&& &= \\begin{pmatrix} \\simplify[std]{{a+c}/sqrt({ssquares})} \\\\ \\simplify[std]{{b+d}/sqrt({ssquares})} \\\\ \\simplify[std]{{g+f}/sqrt({ssquares})} \\end{pmatrix}
\\end{align}
\\begin{align}
\\var{a4}\\boldsymbol{v} &= \\simplify{vector({a4}*{a}, {a4}*{b}, {a4}*{g})} \\\\[1em]
&= \\var{a4*vector(a,b,g)}
\\end{align}
\\begin{align}
\\var{-b4}\\boldsymbol{v} &= \\simplify{vector({-b4}*{c}, {-b4}*{d}, {-b4}*{f})} \\\\[1em]
&= \\var{-b4*vector(c,d,f)}
\\end{align}
Using the information above, the unit vector parallel to $\\boldsymbol{v}$ is:
\n\\[ \\boldsymbol{u_v} = \\begin{pmatrix} \\simplify[std]{{a}/sqrt({ssquaresA})} \\\\ \\simplify[std]{{b}/sqrt({ssquaresA})} \\\\ \\simplify[std]{{g}/sqrt({ssquaresA})} \\end{pmatrix} \\]
\nand the unit vector anti-parallel to $\\boldsymbol{w}$ is:
\n\\[ -\\boldsymbol{u_w} = \\begin{pmatrix} \\simplify[std]{{-c}/sqrt({ssquaresB})} \\\\ \\simplify[std]{{-d}/sqrt({ssquaresB})} \\\\ \\simplify[std]{{-f}/sqrt({ssquaresB})} \\end{pmatrix} \\]
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\n\\begin{align}
\\boldsymbol{v} & =\\simplify[std]{vector({a},{b},{g})}, &
\\boldsymbol{w} &= \\simplify[std]{vector({c},{d},{f})}\\qquad \\in{\\mathbb R}^3.
\\end{align}
Enter your answers to the following questions exactly, using the function sqrt(x) if necessary.