For any vector prove that `vecr=(vecr.veci)i+(vecr.vecj)j+(vecr.veck)k`

For any vector vecr prove that vecr=(vecr.hati)hati+(vecr.hatj)hatj+(vecr.hatk)hatk .

If veca, vecb and vecc be three non-coplanar vectors and a',b' and c' constitute the reciprocal system of vectors, then prove that i. vecr=(vecr.veca')veca+(vecr.vecb')vecb+(vecr.vecc')vecc ii. vecr= (vecr.veca)veca'+(vecr.vecb)vecb' + (vecr.vecc) vecc'

veca , vecb and vecc are three non-coplanar vectors and vecr . Is any arbitrary vector. Prove that [vecbvecc vecr]veca+[vecc veca vecr]vecb+[vecavecbvecr]vecc=[veca vecb vecc]vecr .

If veca,vecb and vecc are three non coplanar vectors and vecr is any vector in space, then (vecxxvecb),(vecrxxvecc)+(vecb xxvecc)xx(vecrxxveca)+(veccxxveca)xx(vecrxxvecb)= (A) [veca vecb vecc] (B) 2[veca vecb vecc]vecr (C) 3[veca vecb vecc]vecr (D) 4[veca vecb vecc]vecr

Let vecr be a non - zero vector satisfying vecr.veca = vecr.vecb =vecr.vecc =0 for given non- zero vectors veca vecb and vecc Statement 1: [ veca - vecb vecb - vecc vecc- veca] =0 Statement 2: [veca vecb vecc] =0

Find (veci+vecj).{(vecj+veck) X (veck+veci)} :

Let vecr, veca, vecb and vecc be four non-zero vectors such that vecr.veca=0, |vecrxxvecb|=|vecr||vecb|,|vecrxxvecc|=|vecr||vecc| then [(veca, vecb, vecc)]=

Let veca=-hati-hatk,vecb =-hati + hatj and vecc =hat i + 2hatj + 3hatk be three given vectors. If vecr is a vector such that vecr xx vecb = vecc xx vecb and vecr.veca =0 then find the value of vecr .vecb .

Find the point of intersection of the line vecr=(vecj-veck)+s(2veci-vecj+veck) and Xz-plane