Let the vectors `veca,vecbvecc` be given as `a_(1)hati+a_(2)hatj+a_(3)hatk,b_(1)hati+b_(2)hatj+b_(3)hatkc_(1)hati+c_(2)hatj+c_(3)hatk`. Then show that `vecaxx(vecb+vecc)=vecaxxvecb+vecaxxvecc`

Let veca=a_(1)hati+a_(2)hatj+a_(3)hatk,vecb=b_(1)hati + b_(2)hatj + b_(3)hatk and vecc=c_(1)hati +c_(2)hatj +c_(3)hatk be three non-zero vectors such that vecc is a unit vector perpendicular to both vectors, veca and vecb . If the angle between veca and vecb is pi//6 "then" |{:(a_(1),a_(2),a_(3)),(b_(1),b_(2),b_(3)),(c_(1),c_(2),c_(3)):}|^(2) is equal to

Let V be the volume of the parallelepied formed by the vectors, veca = a_(1)hati=a_(2)hatj + a_(3) hatk , vecb = b_(1) hati + b_(2)hatj + b_(3) hatk and vecc =c_(1)hati + c_(2)hatj + c_(3)hatk . if a_(r) b_(r) nad c_(r) " where " r= 1,2,3 are non- negative real numbers and sum_(r=1)^(3) (a_(r) + b_(r)+c_(r))=3L " show that " V leL^(3)

Ifveca=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 and [3veca+vecb=vecc 3vecc + veca] =lambda|{:(veca.hati,veca.hatj,veca.hatk),(vecb.hati,veca.hatj,hatb.hatk),(vecc.hati,vecc.hatj,vecc.hatk):}| " then find the value of " lambda/4

Statement 1: If a_(1)hati + a_(2)hatj + a_(3)hatk, b_(1)hati+b_(2)hatj + b_(3) hatk and c_(1)hati + c_(2)hatj + c_(3)hatk are three mutually perpendicular unit vectors then a_(1)hati + b_(1)hatj + c_(1)hatk,a_(2)hati +b_(2)hatj+c_(2) hatk and a_(3)hati + b_(3) hatj + c_(3) hatk may be mutually perpendicular unit vectors. Statement 2 : value of determinant and its transpose are the same.

If veca=2hati+3hatj-hatk, vecb=3hati+5hatj+2hatk, vec c= -hati-2hatj+3hatk , verify that veca xx (vecb xx vec c)=(veca*vec c)vecb-(veca*vecb)vec c

If veca=2hati+3hatj-hatk, vecb=3hati+5hatj+2hatk, vec c= -hati-2hatj+3hatk , verify that (veca xx vecb) xx vec c=(veca *vec c) vecb-(vecb*vec c)veca

If veca=2hati+3hatj-hatk, vecb= -hati+2hatj-4hatk, vec c =hati+hatj+hatk , then find the value of (veca xx vecb)*(veca xx vec c) .