( a =3 i + j - k , b = i -4 j +5 k , c =4 i +5 j - k ) are three vectors and a vector ( r ) is perpendicular to both the vectors ( b ) and ( c ). If ( r a =9 ), then ( r = )

Step-by-step solution

Given vectors ( a =3 i + j - k , b = i -4 j +5 k ) and ( c =4 i +5 j - k ) ( aligned & b c = | array ccc i & j & k 1 & -4 & 5 4 & 5 & -1 array |= i (4-25)- j (- l -20) + k (5+16) &=-21 i +21 j +21 k =21(- i + j + k ) & | b c |=21 3 aligned ) Now, vector ( r ), which is perpendicular to (b ) and ( c ), so ( aligned r = & | r | b c | b c | & r a =9 & | r | | b c | [21(- i + j + k ) (3 i + j - k )]=9 & | r | 21 3 [21(-3+1-1)]=9 & | r |=3 3 ( | r | > 0) & r = 3 3 21(- i + j + k ) 21 3 = 3(- i + j + k ) & =-3( i + j + k