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Some basic tasks involving vectors, including converting to/from component form, scalar product, resultant vectors.

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5 minutes remaining!

"}}, "question_groups": [{"name": "", "pickQuestions": 1, "pickingStrategy": "all-ordered", "questions": [{"name": "Calculate vector magnitude and direction", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Component.gif", "/srv/numbas/media/question-resources/Component.gif"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "variable_groups": [], "variables": {"y": {"description": "", "name": "y", "templateType": "anything", "definition": "random(1..10 except x)", "group": "Ungrouped variables"}, "x": {"description": "", "name": "x", "templateType": "anything", "definition": "random(1..10)", "group": "Ungrouped variables"}, "angle": {"description": "", "name": "angle", "templateType": "anything", "definition": "degrees(arctan(y/x))", "group": "Ungrouped variables"}, "r": {"description": "", "name": "r", "templateType": "anything", "definition": "sqrt(x^2+y^2)", "group": "Ungrouped variables"}}, "ungrouped_variables": ["x", "y", "r", "angle"], "tags": [], "advice": "", "rulesets": {}, "functions": {}, "preamble": {"css": "", "js": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "statement": "

Let $\\underline{v}=\\left(\\begin{array}{c}\\var{x}\\\\ \\var{y}\\end{array}\\right)$

\n

You are going to calculate the magnitude and direction of this vector. Refer to the diagram below:

\n

\n

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What is the horizontal component of $\\underline{v}$?

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What is the vertical component of $\\underline{v}$?

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This is the length of the vector. Refer to Lesson 1 for examples.

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Calculate the magnitude of $\\underline{v}$

\n

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\n

This is the angle $\\theta$ shown in the diagram. Refer to Lesson 1 for examples.

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Calculate the direction of the vector $\\underline{v}$

\n

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A ship travels $\\var{m}$ nautical miles on a bearing of $\\var{d}^\\circ$.

\n

You will calculate its horizontal (east/west) component and its vertical (north/south) component.

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Bearings are measured clockwise from north. Draw a diagram to help you to understand the vector.

\n

The ship has travelled [[0]] and [[1]].

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east

", "

west

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north

", "

south

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East and North will give a positive component.

\n

West and South will give a negative component.

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Calculate the horizontal component of the vector?

\n

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How far east or west has the ship travelled? (If it is west, then enter a negative number.)

\n

Refer to Lesson 1 for examples.

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Calculate the vertical component of the vector?

\n

", "steps": [{"scripts": {}, "prompt": "

How far north or south has the ship travelled? (If it is south, then enter a negative number.)

\n

Refer to Lesson 1 for examples.

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Now write the vector in component form.

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Let $\\underline{a}=\\left(\\begin{array}{c}\\var{x1}\\\\ \\var{y1}\\\\ \\var{z1}\\end{array}\\right)$ and let $\\underline{b}=\\left(\\begin{array}{c}\\var{x2}\\\\ \\var{y2}\\\\ \\var{z2}\\end{array}\\right)$

\n

Calculate the following vectors:

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What is $\\underline{a}+\\underline{b}$?

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What is $\\var{a}\\underline{a}$?

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What is $\\var{b}\\underline{a}-\\var{c}\\underline{b}$?

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Let $A,B,C$ be the points

\n

\$A=(\\var{x1},\\var{y1},\\var{z1})\\qquad B=(\\var{x2},\\var{y2},\\var{z2})\\qquad C=(\\var{x3},\\var{y3},\\var{z3})\$

\n

We will determine whether these points are colinear or not, i.e. do they all lie on a single straight line?

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What is the vector $\\overrightarrow{AB}$?

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What is the vector $\\overrightarrow{BC}$?

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If $\\overrightarrow{BC}$ is a multiple of $\\overrightarrow{AB}$ then the vectors are parallel and the points are colinear.

\n

Otherwise the points are not colinear.

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Yes

", "

No

Are the points colinear?

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Complete the following:

\n

$\\overrightarrow{BC}=$ [[0]]$\\times\\overrightarrow{AB}$

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$|\\underline{a}|$ is the magnitude of vector $\\underline{a}$. Refer to Lesson 1.

", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0}], "prompt": "

What is $|\\underline{a}|$?

\n

", "vsetrange": [0, 1], "checkingaccuracy": "2", "scripts": {}, "marks": 1, "vsetrangepoints": 5}, {"checkingtype": "dp", "variableReplacements": [], "showpreview": false, "type": "jme", "checkvariablenames": false, "expectedvariablenames": [], "answer": "{r2}", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "prompt": "

What is $|\\underline{b}|$?

\n

", "vsetrangepoints": 5, "vsetrange": [0, 1], "checkingaccuracy": "2", "scripts": {}, "marks": 1}, {"checkingtype": "absdiff", "variableReplacements": [], "showCorrectAnswer": true, "type": "jme", "checkvariablenames": false, "expectedvariablenames": [], "answer": "{sp}", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showpreview": false, "stepsPenalty": 0, "steps": [{"variableReplacements": [], "showCorrectAnswer": true, "type": "information", "prompt": "

$\\underline{a}\\cdot\\underline{b}$ is the scalar product of vectors $\\underline{a}$ and $\\underline{b}$. Refer to Lesson 3.

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Calculate $\\underline{a}\\cdot\\underline{b}$

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Rearrange the following formula:

\n

\$\\underline{a}\\cdot\\underline{b}=|\\underline{a}||\\underline{b}|\\cos{\\theta}\$

\n

Refer to Lesson 3 for examples.

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Now calculate the angle between vectors $\\underline{a}$ and $\\underline{b}$.

\n

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Let $\\underline{a}=\\left(\\begin{array}{c}\\var{x1}\\\\ \\var{y1}\\\\ \\var{z1}\\end{array}\\right)$ and let $\\underline{b}=\\left(\\begin{array}{c}\\var{x2}\\\\ \\var{y2}\\\\ \\var{z2}\\end{array}\\right)$

", "preamble": {"css": "", "js": ""}, "ungrouped_variables": ["x1", "y1", "z1", "x2", "y2", "z2", "r1", "r2", "sp", "angle"], "advice": "", "type": "question"}, {"name": "Calculate resultant vector", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Component.gif", "/srv/numbas/media/question-resources/Component.gif"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "advice": "", "rulesets": {}, "ungrouped_variables": ["x2", "y2", "m2", "d2", "x1", "mat2", "mat1", "mr", "dr"], "statement": "

A body is subject to two forces:

\n
\n
• $F_1$ acting horizontally with magnitude $\\var{x1}$ N.
• \n
• $F_2$ acting at $\\var{d2}^\\circ$ to the horizontal with magnitude $\\var{m2}$ N.
• \n
\n

We will calculate the resultant force $F_r$.

\n

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Write the vector $F_1$ in component form.

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Calculate the horizontal component of vector $F_2$

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Calculate the vertical component of vector $F_2$

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Now write vector $F_2$ in component form.

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Calculate the resultant vector $F_r$ of the two vectors $F_1$ and $F_2$.

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The resultant of two vectors is calculated by adding them together. Refer to Lesson 2 for examples.

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Calculate the magnitude of the resultant force $F_r$.

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Calculate the direction of the resultant force $F_r$.