If `vecl,vecm,vecn` are three non coplanar vectors prove that `[vec` vecm vecn](vecaxxvecb) =|(vec1.veca, vec1.vecb, vec1),(vecm.veca, vecm.vecb, vecm),(vecn.veca, vecn.vecb, vecn)|`

Explanation

Let `vecl =l_(1)hati + l_(2)hatj +l_(3)hatk, vecm = m_(1)hati + m_(2)hatj + m_(3)hatk`
` vecn=n_(1)hati+n_(2)hatj +n_(3)hatk, veca = a_(1)hati +a_(2)hatj +a_(3) hatk`
`vecb =b_(1)hati , b_(2)hatj+b_(3)hatk,` therefore,
`vecl.veca =l_(1)a_(1)+ l_(2)a_(2) +l_(3)a_(3) = sum l_(1) a_(1) `
similarly `vecl.vecb = suml_(1)b_(1)`.etc.
`now, [vecl vecm vecn] (vecaxxvecb) = |{:(l_(1),l_(2),l_(3)),(m_(1),m_(2),m_(3)),(n_(1),n_(2),n_(3)):}|xx|{:(hati,hatj ,hatk),(a_(1),a_(2),a_(3)),(b_(1),b_(2),b_(3)):}|`
`|{:(suml_(1)hati,suml_(1)a_(1),suml_(1)b_(1)),(summ_(1)hati,summ_(1)a_(1),summ_(1)b_(1)),(sumn_(1)hati, sumn_(1)a_(1),sumn_(1)b_(1)):}|`
`=|{:(vecl,vecl.veca,vecl.vecb),(vecm,vecm.veca,vecm.vecb),(vecn,vecn.veca,vecn.vecb):}|=|{:(vecl.veca,vecl.vecb,vecl),(vecm.veca,vecm.vecb,vecm),(vecn.veca,vecn.vecb,vecn):}|`