Question

Question #683a7

Answer 1

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`