How will you prove the formula cos(A-B)=cosAcosB+sinAsinB using formula of vector product of two vectors?

1 Answer
P dilip_k
Jun 10, 2016

As below

Explanation:

self draw

Let us consider two unit vectors in X-Y plane as follows :

  • hata-> inclined with positive direction of X-axis at angles A
  • hat b-> inclined with positive direction of X-axis at angles 90+B, where 90+B>A
  • Angle between these two vectors becomes
    theta=90+B-A=90-(A-B),

hata=cosAhati+sinAhatj
hatb=cos(90+B)hati+sin(90+B)
=-sinBhati+cosBhatj
Now
hata xx hatb=(cosAhati+sinAhatj)xx(-sinBhati+cosBhatj)
=>|hata||hatb|sinthetahatk=cosAcosB(hatixxhatj)-sinAsinB(hatjxxhati)
Applying Properties of unit vectos hati,hatj,hatk
hatixxhatj=hatk
hatjxxhati=-hatk
hatixxhati= "null vector"
hatjxxhatj= "null vector"
and
|hata|=1 and|hatb|=1" ""As both are unit vector"

Also inserting
theta=90-(A-B),

Finally we get
=>sin(90-(A-B))hatk=cosAcosBhatk+sinAsinBhatk

:.cos(A-B)=cosAcosB+sinAsinB

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