Voorstelling en notatie

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Gebaseerd op cursus SPC 2021

Verbeteringen/fouten/onduidelijkheden/...?

Deze open-source cursus is in ontwikkeling. De aanbevelingen van leerlingen om dit materiaal te verbeteren zijn erg welkom via info@wiskunde.opmaat.org
Dit kan gaan over:

Een voorbeeld dat onduidelijk is.
Onnauwkeurigheden, schrijffouten, ...
Een tussenstap die beter uitgelegd moet worden.
Een uitleg die je op youtube, wikipedia of aan de kersttafel gevonden hebt die ophelderend was.
...

Vectoren worden grafisch voorgesteld met een pijl. Een vectoriële grootheid wordt genoteerd met $\vec {z}, \vec {a}, \vec {x}, ...$ en wordt altijd bij de pijl gezet ter benoeming. Om duidelijk te maken dat het telkens om een vector gaat wordt een pijltje boven de letter geplaatst. Zonder de vector te benoemen stelt de pijl geen vector voor (en kan het dus evengoed een echte pijl afgeschoten door een boog zijn)! De pijl geeft alle kenmerken die een vector vastleggen weer.

Figuur 1: De vector $\vec {z}$ met zijn kenmerken.

Aan de zin van een vector wordt wiskundig een teken gekoppeld dat afhangt van de gekozen referentie-as. Vectoren in de zin van de gekozen referentie-as worden als positief beschouwd, vectoren tegen de zin van de referentie-as als negatief.

De grootte van een vector $\vec {z}$ wordt aangeduid met de norm $ \lVert \vec {z} \rVert $ of het absolutewaardeteken $\lvert z \rvert $ en is altijd positief. De grootte komt immers overeen met de lengte van de vector (en een lengte is altijd positief).

De richting van een vector wordt weergegeven met een eenheidsvector $\vec {e}$ waarvoor $ \lVert \vec {e} \rVert = 1$. Het invoeren van een eenheidsvector blijkt erg nuttig in notaties. Hiermee kunnen alle kenmerken van een vector ook algebraïsch weergegeven worden.

In onderstaande tekening is $\vec {a} = \pm \lVert \vec {a} \rVert \cdot \vec {e} = +2 \cdot \vec {e} $ met aangrijpingspunt $2$.
Voor de vector $\vec {b}$ geldt dat $\vec {b} = \pm \lVert \vec {b} \rVert \cdot \vec {e} = -1 \cdot \vec {e} $ met aangrijpingspunt $6$.

Figuur 2: Vectoren en aangrijpingspunten.

Als de grootte van een vector $\vec {c}$ gelijk is aan nul, noemt men dit ook de nulvector. Men noteert dit als: $\vec {c} = \vec {0}$ of $ \lVert \vec {c} \rVert $ of $ c = 0 $ Men mag niet noteren dat: $ \vec {c} = 0$ Linkerlid en rechterlid moeten immers beiden een scalar of beiden een vector zijn!

Quick Question 1: Waarom mag je niet noteren dat $\vec {c} = 0$?