General Physics (PHY 170)

• The Dot Product for Vectors -Scalar product • The Cross of Vectors - Vector product

1

Products among VectorsChap 1

Dot product - Scalar product

2

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

3

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

4

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

5

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

6

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

7

In terms of vector components & unit vectors i,j,k are along the x,y,z axes:

Cross or Vector product

Expanding the determinants gives

y

xz

ij

ki

kj

Cyclic: