The vectors are a =2 i + j -2 k , b = i + j . If c is a vector such that a c =| c | and | c - a |=2 2 , angle between a b and c is 4 , then |( a b ) c | is

Step-by-step solution

Given that angle between a b and c is 4 |( a b ) c |=|( a b )|| c | 4 ... (i) Now, a b = | array ccc i & j & k 2 & 1 & -2 1 & 1 & 0 array | aligned & = i (0+2)- j (0+2)+ k (2-1) & =2 i -2 j + k & | a b |= 2^2+(-2)^2+1 =3 aligned Given, a =2 i + j -2 k | a |= 2^2+1^2+(-2)^2 =3 Given, | c - a |=2 2 Squaring on both sides, we get aligned & | c |^2+ - a ^2-2- a c =8 & | c ^2+3^2-2 | c =8 [ a c =| c |] aligned aligned & | c |^2-2| c |+1=0 & (| c |-1)^2=0 & | c |=1 aligned From (i), aligned |( a b ) c | & =|( a b )|| c |