सिद्ध कीजिए कि `hati xx(oversettoaxxhati )+hatj xx(oversettoa xxhatj )+hatk xx(oversettoa xxhatk ) =2oversettoa `

Explanation

हम जानते है कि
` oversetto axx( oversetto b xx oversetto c)=( oversetto a * oversetto c) oversetto b -( oversetto a* oversetto b ) oversetto c`
`therefore hati xx(oversetto axxhati ) =(hati *hati )oversetto a -(hati*oversetto a ) hati =oversetto a -(hati*oversetto a ) hati " "...(1)`
` " "(because hati *hati =1) `
इसी प्रकार `hati xx(oversetto a xxhatj )= oversetto a -(hatj *oversetto a ) hatj " "...(2)`
तथा ` " "hatk xx(oversetto axxhatk) =oversetto a -(hatk *a ) hatk " "...(3) `
समीकरण (1 ) ,(2 ) व (3 ) को जोड़ने पर
` hati xx(oversetto a xxhati )+hatj xx(oversetto a xxhatj )+hatk xx(oversetto axxhatk }...(4)`
माना ` oversetto a =lhati +mhatj +nhatk " "...(5)`
तब ` hati *oversetto a =hati *(lhati+mhatj +nhatk ) `
` " "= l (hati *hati ) + m (hati *hatj )+n(hati *hatk ) `
` " "= l(1) +m (0) +n(0) =1`
इसी प्रकार `hatj *oversetto a= m ` तथा `hatk *oversetto a =n `
अतः समीकरण (5 ) से
` oversetto a = (hati *oversettoa )hati +(hatj *oversettoa ) hatj +(hatk *oversettoa ) hatk " "...(6) `
समीकरण (6 ) का प्रयोग समीकरण (4 ) में करने पर
` hati xx(oversetto axxhati )+hatj xx(oversetto a xxhatj )+ hatk xx(oversetto a xxhatk ) =3oversetto a -oversetto a =2oversetto a `