// Numbas version: exam_results_page_options {"name": "Scalar Product 01", "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questions": [{"name": "01 - Dot Product 01", "extensions": [], "custom_part_types": [], "resources": [["question-resources/dotprod1.png", "/srv/numbas/media/question-resources/dotprod1.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International"}, "statement": "
The dot product or scalar product can be calculated using the following relationship:
\n$ \\underline{a} \\cdot \\underline{b}=|\\underline{a}| |\\underline{b}| \\cos{(\\theta)}$
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\n$ \\underline{a} \\cdot \\underline{b}=$ [[0]] $\\times$ [[1]] $\\times \\cos$([[2]])
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\n$ \\underline{a} \\cdot \\underline{b}=$[[0]]
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\nThe dot product or scalar product can be calculated using the following relationship:
\n$ \\underline{a} \\cdot \\underline{b}=|\\underline{a}| |\\underline{b}| \\cos{(\\theta)}$
\nBecause $\\underline{i}$ is a unit vector, $|\\underline{i}|=$ [[0]]
\nand the angle between $\\underline{i}$ and itself, $\\theta=$ [[1]] degrees
\nSubstitute these values into the formula:
\n$ \\underline{i} \\cdot \\underline{i}=$ [[2]] $\\times$ [[3]] $ \\times \\cos$([[4]])
\n$ \\underline{i} \\cdot \\underline{i}=$[[5]]
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\n$\\underline{j} \\cdot \\underline{j}$,=[[0]]
\nUse the same method to find $\\underline{k} \\cdot \\underline{k}$,
\n$\\underline{k} \\cdot \\underline{k}$,=[[0]]
\n$\\underline{i} \\cdot \\underline{i}=\\underline{j} \\cdot \\underline{j}=\\underline{k} \\cdot \\underline{k}=$ [[0]]
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\nThe dot product or scalar product can be calculated using the following relationship:
\n$ \\underline{a} \\cdot \\underline{b}=|\\underline{a}| |\\underline{b}| \\cos{(\\theta)}$
\nBecause $\\underline{i}$ is a unit vector, $|\\underline{i}|=$ [[0]]
\nBecause $\\underline{j}$ is a unit vector, $|\\underline{j}|=$ [[1]]
\nand the angle between $\\underline{i}$ and $\\underline{j}$, $\\theta=$ [[2]] degrees
\nSubstitute these values into the formula:
\n$ \\underline{i} \\cdot \\underline{j}=$ [[3]] $\\times$ [[4]] $ \\times \\cos$([[5]])
\n$ \\underline{i} \\cdot \\underline{j}=$[[6]]
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\n$\\underline{i} \\cdot \\underline{k}=$[[0]]
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Use the same method to find $\\underline{j} \\cdot \\underline{k}$,
\n$\\underline{j} \\cdot \\underline{k}=$[[0]]
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Generally, whenever any two vectors are perpendicular to each other their scalar product is zero because the angle between the vectors is [[0]]$^{\\circ}$ and $\\cos{(90)}=$[[1]].
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "90", "maxValue": "90", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}]}, {"name": "04 - Dot Product 04", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International"}, "statement": "The formula that we have used, until now, for the Scalr/Dot Product:
\n$\\underline{a} \\cdot \\underline{b}= |\\underline{a}| |\\underline{b}| \\cos{(\\theta)}$
\nis defined for vectors in Polar or Magnitude/Angle form.
\nWe can use the results from the previous sections to create a new formula to do the same thing when our vectors are in Cartesian or Rectangular form.
\nTo keep things a little simpler to begin with, just look at 2, two dimensional vectors:
\nIf $\\underline{a}=a_{1}\\underline{i}+a_{2}\\underline{j}$ and $\\underline{b}=b_{1}\\underline{i}+b_{2}\\underline{j}$
\n", "advice": "", "rulesets": {}, "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Then:
\n$\\underline{a} \\cdot \\underline{b}=(a_{1}\\underline{i}+a_{2}\\underline{j}) \\cdot (b_{1}\\underline{i}+b_{2}\\underline{j})$
\n$\\underline{a} \\cdot\\underline{b}=a_{1}\\underline{i}\\cdot(b_{1}\\underline{i}+b_{2}\\underline{j})+a_{2}\\underline{i}(b_{1}\\underline{i}+b_{2}\\underline{j})$
\n$\\underline{a} \\cdot\\underline{b}=a_{1}b_{1}\\underline{i}\\cdot\\underline{i}+a_{1}b_{2}\\underline{i}\\cdot\\underline{j}+a_{2}b_{1}\\underline{j}\\cdot\\underline{i}+a_{2}b_{2}\\underline{j}\\cdot\\underline{j}$
"}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "This looks daunting, but using the results of our earlier work, we know that:
\n$\\underline{i}\\cdot\\underline{i}=$ [[0]]
\n$\\underline{j}\\cdot\\underline{j}=$ [[1]]
\n$\\underline{i}\\cdot\\underline{j}=$ [[2]]
\nand
\n$\\underline{j}\\cdot\\underline{i}=$ [[3]]
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\nIf $\\underline{a}=a_{1}\\underline{i}+a_{2}\\underline{j}+a_{3}\\underline{k}$ and $\\underline{b}=b_{1}\\underline{i}+b_{2}\\underline{j}+b_{3}\\underline{k}$
\n\n
\n
Using:
\n$\\underline{a} \\cdot\\underline{b}=a_{1}b_{1}+a_{2}b_{2}$
\n\n
Substitute the correct values:
\n$\\underline{a} \\cdot\\underline{b}=$[[0]]$\\times$ [[1]]$\\large{+}$ [[2]]$\\times$ [[3]]
\n\n
Hence, the scalar product is:
\n$\\underline{a} \\cdot\\underline{b}=$[[4]]
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\n
Using:
\n$\\underline{a} \\cdot\\underline{b}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}$
\n\n
Substitute the correct values:
\n$\\underline{a} \\cdot\\underline{b}=$[[0]]$\\times$ [[1]]$\\large{+}$ [[2]]$\\times$ [[3]]$\\large{+}$ [[5]]$\\times$ [[6]]
\n\n
Hence, the scalar product is:
\n$\\underline{a} \\cdot\\underline{b}=$[[4]]
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\n$\\underline{a} \\cdot \\underline{b}=|\\underline{a}| |\\underline{b}| \\cos{(\\theta)}$
\nand
\n$\\underline{a} \\cdot \\underline{b}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}$
\nBoth methods of calculating the scalar product are entirely equivalent and will always give the same value for the scalar product.
\nWe can take advantage of this to calculate the angle between the two vectors. The following example illustrates the method:
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\nUsing:
\n$\\underline{a} \\cdot\\underline{b}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}$
\n\n
Substitute the correct values:
\n$\\underline{a} \\cdot\\underline{b}=$[[0]]$\\times$ [[1]]$\\large{+}$ [[2]]$\\times$ [[3]]$\\large{+}$ [[5]]$\\times$ [[6]]
\n\n
Hence, the scalar product is:
\n$\\underline{a} \\cdot\\underline{b}=$[[4]]
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\n$\\underline{a} \\cdot \\underline{b}=|\\underline{a}| |\\underline{b}| \\cos{(\\theta)}$
\n\nSo we will need to know the values of $|\\underline{a}|$ and $ |\\underline{b}|$. You should remember that:
\n$|\\underline{a}|=\\sqrt{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}$ and $|\\underline{b}|=\\sqrt{b_{1}^{2}+b_{2}^{2}+b_{3}^{2}}$
\nTherefore:
\n$|\\underline{a}|=$ [[0]]
\n$|\\underline{b}|=$ [[1]]
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\n$\\underline{a} \\cdot \\underline{b}=|\\underline{a}| |\\underline{b}| \\cos{(\\theta)}$
\nTo make $\\cos{(\\theta)}$ the subject.
\n\nWhich of these would be a correct rearrangement:
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\nThe angle betwen the two vectors is $\\theta =$ [[0]]
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