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Question
Vectors `veca,vecb and vecc ` are such that `veca+vecb+vecc=0 and |veca| =3,|vecb|=5 and |vecc|=7 ` Find the angle between `veca and vecb`
Solution
We need to find the angle between `veca and vecb`
Now `(veca+vecb+vecc)^2=(veca+vecb+vecc).(veca+vecb+vecc)`
`(veca+vecb+vecc)^2=(|veca|^2+|vecb|^2+|vecc|^2+2veca vecb+2vecc vecb+2veca vecc)`
`0=(9+25+49+2veca vecb+2vecc vecb+2veca vecc)`
`0=83+2(veca vecb+vecc vecb+veca vecc)`
`-83/2=(veca vecb+vecc vecb+veca vecc)`
`-83/2=(veca vecb+(-veca-vecb) vecb+(-veca-vecb) vecc) (because vecc=-veca-vecb)`
`-83/2=(-|vecb|^2-|veca|^2-vecavecb)`
`-83/2=(-|vecb|^2-|veca|^2-|veca||vecb|cos theta)`
`83/2=(34+15cosθ)`
`⇒83/2−34=15cosθ`
`⇒15/2=15cosθ`
`⇒1/2=cosθ`
`θ=π/3 or (5π)/3`
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