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Samenvatting - Wiskunde 'Module 12; analytische meetkunde' GO! Onderwijs $5.39
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Samenvatting - Wiskunde 'Module 12; analytische meetkunde' GO! Onderwijs
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  • Course
  • Wiskunde
  • Institution
  • 2e Graad

Dit document is een samenvatting van 'Module 12; analytische meetkunde', uit het boek 'NANDO 4D' voor het vak Wiskunde in het GO! Onderwijs in de doorstroomfinaliteit/ASO.

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Document information

  • June 24, 2023
  • 3
  • 2022/2023
  • Summary

Subjects

  • wiskunde
  • nando 4
  • module 12
  • analytische meetkunde
  • analytische

Written for

  • Secondary school
  • 2e graad
  • Wiskunde
  • 4
All documents for this subject (69)

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Analytische meetkunde

1. SCALAIR PRODUCT OF INPRODUCT VAN TWEE VECTOREN

1.1 Norm van een vector
Definitie
De norm van een vector ⃗ AB is de grootte van de vector en wordt gelijkgesteld aan de lengte van het
lijnstuk [AB]. -> ‖⃗
AB‖=¿ AB ∨¿
Formule
Als A(x1, y1) en B(x2, y2) dan:
‖⃗AB‖=¿ AB ∨¿ = √ ( x 2−x 1 ) ²+( y 2 − y 1)²
AB‖=√ a +b ²
AB ) = (a, b), dan: ‖⃗
of als co(⃗ 2

Afstandsformule
¿ AB∨¿ = √ ( x 2−x 1 ) ²+( y 2 − y 1)²

1.2 Scalair product of inproduct van twee vectoren
Definitie
Het scalair product van twee vectoren u⃗ en ⃗v is gelijk aan het product van de norm van u⃗ , de norm
van ⃗v en de cosinus van de georiënteerde hoek tussen u⃗ en ⃗v . -> u⃗ ⋅⃗v =‖u⃗‖⋅ ‖⃗v‖⋅ cos ( u^
⃗ , ⃗v )
Eigenschap 1
2
u⃗ ⋅ ⃗u=‖u⃗‖ -> u⃗ ⋅ ⃗u =( u⃗ )2
= ‖u⃗‖⋅ ‖u⃗‖⋅cos 0°
2
= ‖u⃗‖
Eigenschap 2
u⃗ ⊥ ⃗v ⇔ ⃗u ⋅ ⃗v =0 -> u⃗ ⊥ ⃗v ⇨ ⃗u ⋅ ⃗v = ‖u⃗‖⋅ ‖⃗v‖⋅ cos 9 0 °
=0

1.3 Analytische uitdrukking van het inproduct van twee vectoren
Definitie
Het inproduct of scalair product van de vectoren u⃗ (x1, y1) en ⃗v(x2, y2) is de som van het product van
de x-coördinaten en het product van de y-coördinaten. -> u⃗ ⋅ ⃗v =x1 x 2+ y1 y 2

1.4 Hoek tussen twee vectoren
Formule
Als co(u⃗ ) = (x1, y1) en co( ⃗v ) = (x2, y2) dan:
x1 ⋅ x 2+ y 1 ⋅ y 2
cos ( ⃗u^
, ⃗v ) =
Bewijs
√x + y ⋅ √x + y
2
1
2
1
2
2
2
2


Gegeven: u⃗ met coördinaat u⃗ (x1, y1)
⃗v met coördinaat ⃗v (x2, y2)
Oplossing:
1) u⃗ ⋅ ⃗v =‖u⃗ ‖⋅‖⃗v‖⋅ cos ( u^
⃗ , ⃗v )
√
met: ‖u⃗‖= x 21+ y 21
lm ‖u⃗‖= √ x + y 2
2
2
2
2) u⃗ ⋅⃗v =x1 x 2+ y1 y 2
Besluit: ‖u⃗‖⋅ ‖⃗v‖⋅ cos ( u^
⃗ , ⃗v ) ¿ x 1 x 2 + y 1 y 2
x1 ⋅ x 2+ y 1 ⋅ y 2 1
cos ( ⃗u^
, ⃗v ) =
‖u⃗‖⋅ ‖⃗v‖
x1 ⋅ x 2+ y 1 ⋅ y 2

cos ( ⃗u^
, ⃗v ) = 2 2 2 2

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