सदिश विधि द्वारा सिद्ध कीजिए कि
(i) `(veca+vecb)^(2)+veca^(2)+2veca.vecb+vecb^(2)` (ii) `(veca-vecb)^(2)=veca^(2)-2veca.vecb+vecb^(2)`
(iii)`(veca+vecb).(veca-vecb)=veca^(2)-vecb^(2)` (iv) `veca.(vecb-vecc)=veca.vecb-veca.vecc`

Explanation

(i)माना `veca=a_(1)hati+a_(2)hatj+a_(3)hatk`
`vecb=b_(1)hati+b_(2)hatj+b_(3)hatk`
`:.` बायां पक्ष `=(veca+vecb)^(2)`
`=[(a_(1)hati+a_(2)hatj+a_(3)hatk)+(b_(1)hati+b_(2)hatj+b_(3)hatk)]^(2)`
`(a_(1)hati+a_(2)hatj+a_(3)hatk)^(2)+(b_(1)hati+a_(2)hatj+a)_(3)hatk).(b_(1)hati+b_(2)hatj+b_(3)hatk)`
`[ :' veca_(2)=a_(1)^(2)+a_(2)^(2) +a_(3)^(2), vecb_(2)=b_(1)^(2)+b_(2)^(2)+b_(3)^(2)]`
`=(a_(1)^(2)+a_(2)^(2)+a_(3)^(2))+(b_(1)^(2)+b_(2)^(2)+b_(3)^(2))+2(a_(1)b_(1)+b_(1)b_(2)+a_(3)b_(3))`
`=veca^(2)+vecb^(2)+2veca.vecb`
`veca^(2)+2veca.vecb+vecb^(2)`
(ii) `(veca-vecb)^(2)=veca^(2)-2veca.vecb+vecb^(2)`
बायां पक्ष `=(veca-vecb)^(2)`
`=[(veca_(1)hati+a_(2)hatj+a_(3)hatk)-(b_(1)hati+b_(2)hatj+b_(3)hatk)]^(2)`
`=(a_(1)hati+a_(2)hatj+a_(3)hatk)^(2)+(b_(1)hati+b_(2)hatj+b_(3)hatk)^(2)`
`-2(a_(1)hati+a_(2)hatj+a_(3)hatk).(b_(1)hati+b_(2)hatj+b_(3)hatk)`
`=(a_(1)^(2)+a_(2)^(2)+a_(3)^(2))+(b_(1)^(2)+b_(2)^(2)+ b_(3)^(2))-2(a_(1)b_(1)+a_(2)b_(2)+a_(3)b_(3))`
`=veca_(2)-2veca.vecb+vecb_(2)`
(iii) `(veca+vecb).(veca-vecb)=veca^(2)-vecb^(2)`
बायां पक्ष `=(veca+veca-vecb)`
`={(a_(1)hati+a_(2)hatj+a_(3)hatk)+(b_(1)hati+b_(2)hatk)}. {(a_(1)hati+a_(2)hatj+a_(3)hatk-b_(1)hati-b_(2)hatj-b_(3)hatk)}`
`=(a_(1)+b_(1))hati+(a_(2)+b_(2))hatj+(a_(3)+b_(3))hatk}.{(a_(1)-b_(1)hati+(a_(@)-b_(2))hatj+(a_(3)-b_(3))hatk}`
`=(a_(1)+b_(1))(a_(1)-b_(1))+(a_(2)+b_(2))(a_(2)-b_(2))+(a_(3)+b_(3))(a_(3)-b_(3))`
`=veca_(2)-vecb_(2)`
(iv) `veca.(vecb-vecc)=veca.vecb-veca.vecc`
माना `veca=a_(1)hati+a_(2)hatj+a_(3)hatk`
`vecb=b_(1)hati+b_(2)hatj+b_(3)hatk`
`vecc=c_(1)hati+c_(2)hatj+c_(3)hatk`
बायां पक्ष `veca.(vecb-vecc)`
`=(a_(1)hati+a_(2)hatj+a_(3)hatk).[(b_(1)hati+b_(2)hatj+b_(3)hatk)-(c_(1)hati+c_(2)hatj+c_(3)hatk)]`
`=(a_(1)hati+a_(2)hatj+a_(3)hatk).[(b_(1)-c_(1))hati+(b_(2)-c_(2))hatj+(b_(3)-c_(3))hatk]`
`=a_(1)(b_(1)-c_(1))+a_(2)(b_(2)-c_(2))+a_(3)(b_(3)-c_(3))`
`=a_(1)b_(1)-a_(1)c_(1)+a_(2)b_(2)-a_(2)c_(2)+a_(2)b_(2)+a_(3)C_(3)`
`=a_(1)b_(1)+a_(2)b_(2)+a_(3)b_(3)-(a_(1)c_(1)+a_(2)c_(2)+a_(3)c_(3))`
`=veca.vecb-veca.vecc`