Volumen von Tetraeder
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
Short
Video
\(\LaTeX\)
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Don't forget to subscribe to our channel, like the videos and leave comments!
Exercise:
Ein Tetraeder ist im Raum durch die Punkte AAxAyAz BBxByBz CCxCyCz DDxDyDz definiert. Welches Volumen hat er?
Solution:
center tikzpicturescale. pgfmathsetmacrofactor/sqrt; coordinate labelright:AAx Ay Az A at AxAyAz*factor; coordinate labelleft:BBx By Bz B at BxByBz*factor; coordinate labelabove:CCx Cy Cz C at CxCyCz*factor; coordinate labelbelow:DDx Dy Dz D at DxDyDz*factor; draw- -- noderight x; draw- -- nodeabove y; draw- -- nodebelow left z; draw- opacity. A--D--B--cycle; draw- opacity. A --D--C--cycle; draw- opacity. B--D--C--cycle; tikzpicture center pgfmathsetmacroarootax^+ay^+az^ pgfmathsetmacroaaroot^. pgfmathsetmacroV^.*a^/ Das Volumen eines Tetraeders kann mit der allgemeinen Formel für ein Spat ausgerechnet werden. V fracvmatrixdetpmatrixvecavecbveccpmatrixvmatrix Für diese Formel muss man jedoch noch die Richtungsvektoren von einem Beliebigen Punkt aus ausrechnen. In diesem Fall vom Punkt A. veca vecB-vecA pmatrixnumround-pad falseax numround-pad falseay numround-pad falseazpmatrix vecb vecD-vecA pmatrixnumround-pad falsebx numround-pad falseby numround-pad falsebzpmatrix vecc vecC-vecA pmatrixnumround-pad falsecx numround-pad falsecy numround-pad falseczpmatrix Jetzt muss man nur noch die Vektoren in die Formel einsetzen. V fracvmatrixveca times vecbveccvmatrix % fracvmatrixpmatrixaybz-azby azbx-axbz axby-aybxpmatrixpmatrixcx cy czpmatrixvmatrix fracvmatrixpmatrixnumround-pad falsecrossx numround-pad falsecrossy numround-pad falsecrosszpmatrixpmatrix numround-pad falsecx numround-pad falsecy numround-pad falseczpmatrix vmatrix fracvmatrixpmatrixvx vy vzpmatrixvmatrix fracsqrtnegSquarevx+negSquarevy+negSquarevz fracsqrtnumround-pad falsesqroot fracnumround-pad falseroot V numround-pad falseresult
Ein Tetraeder ist im Raum durch die Punkte AAxAyAz BBxByBz CCxCyCz DDxDyDz definiert. Welches Volumen hat er?
Solution:
center tikzpicturescale. pgfmathsetmacrofactor/sqrt; coordinate labelright:AAx Ay Az A at AxAyAz*factor; coordinate labelleft:BBx By Bz B at BxByBz*factor; coordinate labelabove:CCx Cy Cz C at CxCyCz*factor; coordinate labelbelow:DDx Dy Dz D at DxDyDz*factor; draw- -- noderight x; draw- -- nodeabove y; draw- -- nodebelow left z; draw- opacity. A--D--B--cycle; draw- opacity. A --D--C--cycle; draw- opacity. B--D--C--cycle; tikzpicture center pgfmathsetmacroarootax^+ay^+az^ pgfmathsetmacroaaroot^. pgfmathsetmacroV^.*a^/ Das Volumen eines Tetraeders kann mit der allgemeinen Formel für ein Spat ausgerechnet werden. V fracvmatrixdetpmatrixvecavecbveccpmatrixvmatrix Für diese Formel muss man jedoch noch die Richtungsvektoren von einem Beliebigen Punkt aus ausrechnen. In diesem Fall vom Punkt A. veca vecB-vecA pmatrixnumround-pad falseax numround-pad falseay numround-pad falseazpmatrix vecb vecD-vecA pmatrixnumround-pad falsebx numround-pad falseby numround-pad falsebzpmatrix vecc vecC-vecA pmatrixnumround-pad falsecx numround-pad falsecy numround-pad falseczpmatrix Jetzt muss man nur noch die Vektoren in die Formel einsetzen. V fracvmatrixveca times vecbveccvmatrix % fracvmatrixpmatrixaybz-azby azbx-axbz axby-aybxpmatrixpmatrixcx cy czpmatrixvmatrix fracvmatrixpmatrixnumround-pad falsecrossx numround-pad falsecrossy numround-pad falsecrosszpmatrixpmatrix numround-pad falsecx numround-pad falsecy numround-pad falseczpmatrix vmatrix fracvmatrixpmatrixvx vy vzpmatrixvmatrix fracsqrtnegSquarevx+negSquarevy+negSquarevz fracsqrtnumround-pad falsesqroot fracnumround-pad falseroot V numround-pad falseresult
Meta Information
Exercise:
Ein Tetraeder ist im Raum durch die Punkte AAxAyAz BBxByBz CCxCyCz DDxDyDz definiert. Welches Volumen hat er?
Solution:
center tikzpicturescale. pgfmathsetmacrofactor/sqrt; coordinate labelright:AAx Ay Az A at AxAyAz*factor; coordinate labelleft:BBx By Bz B at BxByBz*factor; coordinate labelabove:CCx Cy Cz C at CxCyCz*factor; coordinate labelbelow:DDx Dy Dz D at DxDyDz*factor; draw- -- noderight x; draw- -- nodeabove y; draw- -- nodebelow left z; draw- opacity. A--D--B--cycle; draw- opacity. A --D--C--cycle; draw- opacity. B--D--C--cycle; tikzpicture center pgfmathsetmacroarootax^+ay^+az^ pgfmathsetmacroaaroot^. pgfmathsetmacroV^.*a^/ Das Volumen eines Tetraeders kann mit der allgemeinen Formel für ein Spat ausgerechnet werden. V fracvmatrixdetpmatrixvecavecbveccpmatrixvmatrix Für diese Formel muss man jedoch noch die Richtungsvektoren von einem Beliebigen Punkt aus ausrechnen. In diesem Fall vom Punkt A. veca vecB-vecA pmatrixnumround-pad falseax numround-pad falseay numround-pad falseazpmatrix vecb vecD-vecA pmatrixnumround-pad falsebx numround-pad falseby numround-pad falsebzpmatrix vecc vecC-vecA pmatrixnumround-pad falsecx numround-pad falsecy numround-pad falseczpmatrix Jetzt muss man nur noch die Vektoren in die Formel einsetzen. V fracvmatrixveca times vecbveccvmatrix % fracvmatrixpmatrixaybz-azby azbx-axbz axby-aybxpmatrixpmatrixcx cy czpmatrixvmatrix fracvmatrixpmatrixnumround-pad falsecrossx numround-pad falsecrossy numround-pad falsecrosszpmatrixpmatrix numround-pad falsecx numround-pad falsecy numround-pad falseczpmatrix vmatrix fracvmatrixpmatrixvx vy vzpmatrixvmatrix fracsqrtnegSquarevx+negSquarevy+negSquarevz fracsqrtnumround-pad falsesqroot fracnumround-pad falseroot V numround-pad falseresult
Ein Tetraeder ist im Raum durch die Punkte AAxAyAz BBxByBz CCxCyCz DDxDyDz definiert. Welches Volumen hat er?
Solution:
center tikzpicturescale. pgfmathsetmacrofactor/sqrt; coordinate labelright:AAx Ay Az A at AxAyAz*factor; coordinate labelleft:BBx By Bz B at BxByBz*factor; coordinate labelabove:CCx Cy Cz C at CxCyCz*factor; coordinate labelbelow:DDx Dy Dz D at DxDyDz*factor; draw- -- noderight x; draw- -- nodeabove y; draw- -- nodebelow left z; draw- opacity. A--D--B--cycle; draw- opacity. A --D--C--cycle; draw- opacity. B--D--C--cycle; tikzpicture center pgfmathsetmacroarootax^+ay^+az^ pgfmathsetmacroaaroot^. pgfmathsetmacroV^.*a^/ Das Volumen eines Tetraeders kann mit der allgemeinen Formel für ein Spat ausgerechnet werden. V fracvmatrixdetpmatrixvecavecbveccpmatrixvmatrix Für diese Formel muss man jedoch noch die Richtungsvektoren von einem Beliebigen Punkt aus ausrechnen. In diesem Fall vom Punkt A. veca vecB-vecA pmatrixnumround-pad falseax numround-pad falseay numround-pad falseazpmatrix vecb vecD-vecA pmatrixnumround-pad falsebx numround-pad falseby numround-pad falsebzpmatrix vecc vecC-vecA pmatrixnumround-pad falsecx numround-pad falsecy numround-pad falseczpmatrix Jetzt muss man nur noch die Vektoren in die Formel einsetzen. V fracvmatrixveca times vecbveccvmatrix % fracvmatrixpmatrixaybz-azby azbx-axbz axby-aybxpmatrixpmatrixcx cy czpmatrixvmatrix fracvmatrixpmatrixnumround-pad falsecrossx numround-pad falsecrossy numround-pad falsecrosszpmatrixpmatrix numround-pad falsecx numround-pad falsecy numround-pad falseczpmatrix vmatrix fracvmatrixpmatrixvx vy vzpmatrixvmatrix fracsqrtnegSquarevx+negSquarevy+negSquarevz fracsqrtnumround-pad falsesqroot fracnumround-pad falseroot V numround-pad falseresult
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