Gerade durch Ebene
About points...
We associate a certain number of points with each exercise.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
About difficulty...
We associate a certain difficulty with each exercise.
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
Question
Solution
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Video
\(\LaTeX\)
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Exercise:
In welchem Punkt durchstösst die Gerade g: pmatrix x y z pmatrix pmatrix Px Py Pz pmatrix + t pmatrix tx ty tz pmatrix die Ebene mathcalE: Ex x + Ey y + Ez z Eih?
Solution:
tdplotsetmaincoords center tikzpicturelatex scale. tdplot_main_coords tikzsetscaled unit vectors. foreach x in --... drawcolorgray scaled cs x---x; foreach y in --... drawcolorgray scaled cs -y--y; drawcolorgreen!!black- scaled cs --- noderight small bmx; drawcolorgreen!!black- scaled cs --- nodeabove small bmy; drawcolorgreen!!black- scaled cs --- nodeleft small bmz; drawdotted scaled cs PxXPyXPzX--PxXPyX; drawdotted scaled cs ----; drawcolorblue scaled cs PxXPyXPzX--+-*txX-*tyX-*tzX; drawcolorblue scaled cs PxXPyXPzX--+*txX*tyX*tzX; draw- stealth colorred thick scaled cs PxXPyXPzX--+txXtyXtzX nodemidwayright tiny vec vpmatrix tx ty tzpmatrix; filldrawcolorblack fillyellow!!white opacity. scaled cs AX--BX--CX--cycle; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesPxXPyXPzX nodeleft tiny PPx|Py|Pz; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesAX noderight tiny A; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesBX nodeleft tiny B; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesCX nodeabove right tiny C; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesDxXDyXDzX nodeleft tiny D; tikzpicture center Lösungsidee: bf einsetzen! center einsetzen in mathcalE cases x -+t y -+t z -t cases center -+t--+t+-t -+t--+t+-t- -+t+-t+-t- -t t t tX Dieses t wird nun bei den oberen drei Komponentengleichungen eingesetzt so erhält man schliesslich den Durchstosspunkt D. center t einsetzen cases x -+ tX y -+ tX z - tX cases center Das führt dann schliesslich auf den Durchstosspunkt D|-|.
In welchem Punkt durchstösst die Gerade g: pmatrix x y z pmatrix pmatrix Px Py Pz pmatrix + t pmatrix tx ty tz pmatrix die Ebene mathcalE: Ex x + Ey y + Ez z Eih?
Solution:
tdplotsetmaincoords center tikzpicturelatex scale. tdplot_main_coords tikzsetscaled unit vectors. foreach x in --... drawcolorgray scaled cs x---x; foreach y in --... drawcolorgray scaled cs -y--y; drawcolorgreen!!black- scaled cs --- noderight small bmx; drawcolorgreen!!black- scaled cs --- nodeabove small bmy; drawcolorgreen!!black- scaled cs --- nodeleft small bmz; drawdotted scaled cs PxXPyXPzX--PxXPyX; drawdotted scaled cs ----; drawcolorblue scaled cs PxXPyXPzX--+-*txX-*tyX-*tzX; drawcolorblue scaled cs PxXPyXPzX--+*txX*tyX*tzX; draw- stealth colorred thick scaled cs PxXPyXPzX--+txXtyXtzX nodemidwayright tiny vec vpmatrix tx ty tzpmatrix; filldrawcolorblack fillyellow!!white opacity. scaled cs AX--BX--CX--cycle; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesPxXPyXPzX nodeleft tiny PPx|Py|Pz; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesAX noderight tiny A; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesBX nodeleft tiny B; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesCX nodeabove right tiny C; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesDxXDyXDzX nodeleft tiny D; tikzpicture center Lösungsidee: bf einsetzen! center einsetzen in mathcalE cases x -+t y -+t z -t cases center -+t--+t+-t -+t--+t+-t- -+t+-t+-t- -t t t tX Dieses t wird nun bei den oberen drei Komponentengleichungen eingesetzt so erhält man schliesslich den Durchstosspunkt D. center t einsetzen cases x -+ tX y -+ tX z - tX cases center Das führt dann schliesslich auf den Durchstosspunkt D|-|.
Meta Information
Exercise:
In welchem Punkt durchstösst die Gerade g: pmatrix x y z pmatrix pmatrix Px Py Pz pmatrix + t pmatrix tx ty tz pmatrix die Ebene mathcalE: Ex x + Ey y + Ez z Eih?
Solution:
tdplotsetmaincoords center tikzpicturelatex scale. tdplot_main_coords tikzsetscaled unit vectors. foreach x in --... drawcolorgray scaled cs x---x; foreach y in --... drawcolorgray scaled cs -y--y; drawcolorgreen!!black- scaled cs --- noderight small bmx; drawcolorgreen!!black- scaled cs --- nodeabove small bmy; drawcolorgreen!!black- scaled cs --- nodeleft small bmz; drawdotted scaled cs PxXPyXPzX--PxXPyX; drawdotted scaled cs ----; drawcolorblue scaled cs PxXPyXPzX--+-*txX-*tyX-*tzX; drawcolorblue scaled cs PxXPyXPzX--+*txX*tyX*tzX; draw- stealth colorred thick scaled cs PxXPyXPzX--+txXtyXtzX nodemidwayright tiny vec vpmatrix tx ty tzpmatrix; filldrawcolorblack fillyellow!!white opacity. scaled cs AX--BX--CX--cycle; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesPxXPyXPzX nodeleft tiny PPx|Py|Pz; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesAX noderight tiny A; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesBX nodeleft tiny B; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesCX nodeabove right tiny C; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesDxXDyXDzX nodeleft tiny D; tikzpicture center Lösungsidee: bf einsetzen! center einsetzen in mathcalE cases x -+t y -+t z -t cases center -+t--+t+-t -+t--+t+-t- -+t+-t+-t- -t t t tX Dieses t wird nun bei den oberen drei Komponentengleichungen eingesetzt so erhält man schliesslich den Durchstosspunkt D. center t einsetzen cases x -+ tX y -+ tX z - tX cases center Das führt dann schliesslich auf den Durchstosspunkt D|-|.
In welchem Punkt durchstösst die Gerade g: pmatrix x y z pmatrix pmatrix Px Py Pz pmatrix + t pmatrix tx ty tz pmatrix die Ebene mathcalE: Ex x + Ey y + Ez z Eih?
Solution:
tdplotsetmaincoords center tikzpicturelatex scale. tdplot_main_coords tikzsetscaled unit vectors. foreach x in --... drawcolorgray scaled cs x---x; foreach y in --... drawcolorgray scaled cs -y--y; drawcolorgreen!!black- scaled cs --- noderight small bmx; drawcolorgreen!!black- scaled cs --- nodeabove small bmy; drawcolorgreen!!black- scaled cs --- nodeleft small bmz; drawdotted scaled cs PxXPyXPzX--PxXPyX; drawdotted scaled cs ----; drawcolorblue scaled cs PxXPyXPzX--+-*txX-*tyX-*tzX; drawcolorblue scaled cs PxXPyXPzX--+*txX*tyX*tzX; draw- stealth colorred thick scaled cs PxXPyXPzX--+txXtyXtzX nodemidwayright tiny vec vpmatrix tx ty tzpmatrix; filldrawcolorblack fillyellow!!white opacity. scaled cs AX--BX--CX--cycle; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesPxXPyXPzX nodeleft tiny PPx|Py|Pz; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesAX noderight tiny A; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesBX nodeleft tiny B; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesCX nodeabove right tiny C; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesDxXDyXDzX nodeleft tiny D; tikzpicture center Lösungsidee: bf einsetzen! center einsetzen in mathcalE cases x -+t y -+t z -t cases center -+t--+t+-t -+t--+t+-t- -+t+-t+-t- -t t t tX Dieses t wird nun bei den oberen drei Komponentengleichungen eingesetzt so erhält man schliesslich den Durchstosspunkt D. center t einsetzen cases x -+ tX y -+ tX z - tX cases center Das führt dann schliesslich auf den Durchstosspunkt D|-|.
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