Let a =2 i + j -2 k and b = i + j be two vectors. c is a vector such that a c =| c | and | c - a |=2 2 . If the angle between a b and c is 30^ , then |( a b ) c | is equal to

Step-by-step solution

aligned & Let a =2 i + j -2 k & aligned | a | & = 2^2+1^2+(-2)^2 =3 and b = i + j | b | & = 1+1 = 2 a b & = | array ccc i & j & k 2 & 1 & -2 1 & 1 & 0 array |= i (2)- j (2)+ k (1)=2 i -2 j + k | a b | & = 2^2+(-2)^2+(1)^2 =3 aligned aligned Now, a c =| c | and | c - a |=2 2 Squaring both side gathered | c - a |^2=(2 2 )^2 | c ^2 |+ | a ^2 |-2 c a =8 | c |^2+9-2| c |=8 | c |^2-2| c |+1=0 gathered Let | c |=x aligned & x^2-2 x+1=0 (x-1)^2=0 & x=1 | c |=1 & Now, |( a b ) c |=| a b || c | 30^ & & aligned