2.14.1 Examples: Scalar product

Let

a=(a1an)Rb=(b1bn)R

be two vectors with real components:

Definition 29 ab=a1b1++aNbN=j=1kajbj is called scalar product of a and b
Note: a,bRN,abR!!

Example:

a=(1234),b=(5678)ab=5+12+21+32=70a=(111),b=(110)ab=11+0=0

Example from physics:
Work:

PIC
W=Fs=|F||s|cosφ
cos(a,b)=ab|a||b|ab=0ifa=0orb=0orab}in 3D

now definitions for a,bRN

  1. a,b0. If ab=0 then a is called orthogonal or perpendicular to b or ab

  2. ab|a||b|1. We define cosφ=ab|a||b| where φ is the angle between the two vectors a and b in R

Example:

  1. a=(111)b=(220)cosφ=2+2+038=23φ=0.615=35.26
  2. a=(1111)b=(2200)cosφ=2+2+0+048=12φ=π4=45a=(1111)b=(1111)ab=111+1=0ab


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© J. Carstensen (Math for MS)