`vecA=(2veci+veck),vecB=(veci+vecj+veck) and vecC=4veci-vec3j+7veck` determine a vector `verR` satisfying `vecRxxvecB=vecCxxvecB and vecR.vecA=0`

Answer

`-hati-8hatj+2hatk`

Explanation

we are given that `vecA=2hati + hatk,vecB=hati+hatj+hatk and vecC= 4hati -3hatj +7hatk and ` to determine a vector `vecR` such that `vecR xx vecB = vecC xx vecB and vecR.vecA =0` Let `vecR =x hati + yhatj + zhatk`
then `vceR xx vecB = vecC xx vecB`
`Rightarrow |{:(hati,hatj,hatk),(x,y,z),(1,1,1):}|=|{:(hati,hatj,hatk),(4,-3,7),(1,1,1):}|`
`or (y-z) hati - (x-z) hatj + (x-y) hatk`
` =-10 hati + (x -z) hatj + 7 hatk`
y-z= -10
x-z =-3
x-y= 7
Also ` vecR.vecA=0`
` Rightarrow 2x +z=0`
Subsituting y =x-7 and z =-2x from (ii) and (iv), respectively in (i) , we get
x-7 +2x =-10
3x=-3
x=-1,y =-8 and z=2