सदिश`3hati + 2hatj -hatk "तथा" 12 hati + 5hatj - 5hatk ` में से प्रत्येक पर लम्ब इकाई सदिश ज्ञात करे |

Explanation

माना की `veca = 3hati + 2hatj - 5hatk "तथा" vecb = 12 hati + 5hatj - 5hatk `
माना की सदिश ` vecc = c_(1) hati + c_(2) hatj + c_(3) hatk, veca "तथा" vecb ` में से प्रत्येक पर लम्ब है
तो `vecc bot veca rArr vecc. Veca = 0 `
` rArr (c_(1)hati + c_(2) hatj + c_(3)hatk).(3hati + 2hatj - 5hatk)`
` rArr 3c_(1) + 2c_(2) -c_(2) = 0` ...(1)
पुनः `vecc bot vecb rArr vecc. vecb = 0 `
` rArr (c_(1)hati + c_(2) hatj + c_(3)hatk).(12hati + 5hatj - 5hatk)=0`
`rArr `12c_(1) + 5c_(2) - 5c_(3) = 0` ...(2)
समीकरण (1) और (2) को हल करने पर हमे मिलता है ,
`(c_(1))/(-10=5) = (c_(2))/(-12+ 15)= (c_(3))/(15-24) " या "(c_(1))/(-5) = (c_(2))/(3) = (c_(3))/(-9) = lambda ` (माना )
` therefore c_(1) = - 5lambda, c_(2) = 3lambda , c_(3) = -9lambda `
` therefore vecc = c_(1) hati + c_(2) hatj + c_(3) hatk = lambda [-5hati + 3hatj - 9hatk]`
`therefore |vecc|^(2) = lambda ^(2) (25 + 9+ 81) rArr |vecc| = |lambda |sqrt(115)`
` therefore = (vecc)/(|vecc|) = (lambda (-5hati + 3hatj- 9hatk))/(|lambda|sqrt(115))`
` = pm (1)/(sqrt(115)) (-5hati + 3hatj - 9hatk )" "[because |lambda| = pm lambda]`
अतः दिय हुए शत्तों को संतुष्ट हुए एक इकाई सदिश
` = (1)/(sqrt(115)) (-5hati + 3hatj - 9hatk)`