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If theta is the angle between the unit v...

If `theta` is the angle between the unit vectors `veca and vecb`, then prove that
i. `cos((theta)/2)=1/2|veca+vecb|`
` ii. sin (theta/2)=1/2|veca-vecb|`

Text Solution

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i. `(veca+vecb).(veca+vecb)= |veca|^(2)+|vecb|^(2)+2veca.vecb`
`=1 +1+2(1)(1) cos theta`
`cos(theta/2)=1/2|veca+vecb|`
ii. `(veca-vecb).(veca-vecb)=|veca|^(2)=|vecb|^(2)-2veca.vecb`
`= 1+1-2(1)(1)cos theta`
`=2-2costheta`
`or sin"theta/2=1/2|veca+vecb|`
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