Let a =2 i - j + k , b = i +2 j - k and c = i + j -2 k be three vectors. A vector of the type b + c for some scalar , whose projection on a is of magnitude 2 3 is :

Step-by-step solution

Let d = b + c aligned & d = i +2 j - k + ( i + j -2 k ) & =(1+ ) i +(2+ ) j -(1+2 ) k aligned If be the angle between d and a , then projection of d or ( b + c ) on a aligned & =| d | & =| d | ( d a | d || a | )= d a | a | & = 2( +1)-( +2)-(2 +1) 4+1+1 & = - -1 6 aligned But projection of d on a = 2 3 - +1 6 = 2 3 ^2+2 +1 6 = 2 3 aligned & ^2+2 -3=0 ^2+3 - -3=0 & ( +3)-1( +3)=0, =1,-3 aligned when =1 , then b + c =2 i +3 j -3 k when =-3 , then b + c =-2 i - j +5 k