If the magnitude of the vector product of the vector ( i + j + k ) with a unit vector along the sum of the vectors (2 i +4 j -5 k ) and ( i +2 j +3 k ) is equal to ( 2 ), then the value of ' ( ) ' is

Step-by-step solution

( aligned Let a & = i + j + k b & =2 i +4 j -5 k c & = i +2 j +3 k b + c & =(2+ ) i +6 j -2 k aligned ) ( d = ) unit vector along (( b + c )= b + c | b + c | ) ( aligned = & (2+ ) i +6 j -2 k (2+ )^2+36+4 d & = (2+ ) i +6 j -2 k ^2+4 +44 aligned ) ( aligned & a d & = | array ccc i & j & k 1 & 1 & 1 (2+ ) ^2+4 +44 & 6 ^2+4 +44 & -2 ^2+4 +44 array | aligned ) ( aligned & = 1 ^2+4 +44 | array ccc i & j & k 1 & 1 & 1 2+ & 6 & -2 array | & = 1 ^2+4 +44 | i (-8)- j (-2-2- )+ k (6-2- )| & a d = 1 ^2+4 +44 |-8 i +(4+ ) j +