chap 1 products among vectors - university of hawaiʻi
TRANSCRIPT
General Physics (PHY 170)
• The Dot Product for Vectors -Scalar product • The Cross of Vectors - Vector product
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Products among VectorsChap 1
Dot product - Scalar product
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The Dot or Scalar Product of two vectors is defined as: A•B ≡ AB cosθ is a scalar!
whereθis the angle between A and B
(A+B)•C = A•C + B•C
Geometric meaning
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A•B ≡ AB cosθ
means A times the projection (B cosθ) of B on A.
A•B = 0 for A =0 or B =0 or A⊥B (cos 90o=0)A•B = AB for A || B (cos 0o=1)A•A = A2
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In terms of vector components & unit vectors i,j,k are along the x,y,z axes:
A = Axi + Ayj + Azk B = Bxi + Byj + Bzk
i•i = j•j = k•k = 1x1x cos0o = 1, i•j= i•k = j•k = 1x1x cos90o = 0,
A•B = (Axi + Ayj + Azk)•(Bxi + Byj + Bzk) = = AxBx + AyBy + AzBz is a scalar!
Dot product - Scalar product
y
xz
ij
k
Cross or Vector product
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If A & B are vectors: C = AXB (read “A cross B”) C is a vector! Module: |C|=AB sinθDirection: perpendicular to both A and B and determined by the right-hand rule
Two nonzero vectors A and B are parallel
(θ=0) when:
A X B = 0
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Cross or Vector product
the order in which the vectors are multiplied is important!
A X B = -B X A
✦Place A and B tail to tail✦Right hand, not left hand✦Four fingers are pointed along the first vector A“sweep” from first vector A into second vector B through the smaller angle between them.✦Your outstretched thumb points the direction of C
Right hand rule