Find the unit vector parallel to the vector i ^ + 2 j ^ + 3 k ^ .

Unit vector corresponding to the vector a → = i ^ + 2 j ^ + 3 k ^ is given by a → a → = i ^ + 2 j ^ + 3 k ^ 1 2 + 2 2 + 3 2 = 1 1 + 4 + 9 i ^ + 2 j ^ + 3 k ^ , which is of the form k a → where k = 1 14 . . . . ( 1 ) We know that a scalar multiple of r → is always parallel to r → . . . ( 2 ) Thus, the unit vector parallel to a → = i ^ + 2 j ^ + 3 k ^ is 1 14 i ^ + 2 j ^ + 3 k ^ . . . (From ( 1 ) and ( 2 ) )

Find the unit vector parallel to the vector i ^ + 2 j ^ + 3 k ^ .

Unit vector corresponding to the vector a → = i ^ + 2 j ^ + 3 k ^ is given by a → a → = i ^ + 2 j ^ + 3 k ^ 1 2 + 2 2 + 3 2 = 1 1 + 4 + 9 i ^ + 2 j ^ + 3 k ^ , which is of the form k a → where k = 1 14 . . . . ( 1 ) We know that a scalar multiple of r → is always parallel to r → . . . ( 2 ) Thus, the unit vector parallel to a → = i ^ + 2 j ^ + 3 k ^ is 1 14 i ^ + 2 j ^ + 3 k ^ . . . (From ( 1 ) and ( 2 ) )

If a → = 3 i ^ - 4 j ^ + 12 k ^ , find a vector b → which is parallel to a → and the direction is opposite to that of a → and with magnitude 39 unit.

Given that a → = 3 i ^ - 4 j ^ + 12 k ^ . Vector b → is parallel to a → and has the direction opposite to that of a → . Then, b → = - c a → where c is a scalar which is positive. ⇒ b → = - c ( 3 i ^ - 4 j ^ + 12 k ) ^ ⇒ b → = - 3 c i ^ + 4 c j ^ - 12 c k ^ Now, b → = ( - 3 c ) 2 + ( 4 c ) 2 + ( - 12 c ) 2 = 39 ⇒ 9 c 2 + 16 c 2 + 144 c 2 = 39 ⇒ 169 c 2 = 39 ⇒ 13 c = 39 ( ∵ c > 0 ) ⇒ c = 3 Thus, we get b → = - 3 ( 3 i ^ - 4 j ^ + 12 k ^ ) = - 9 i ^ + 12 j ^ - 36 k ^ , which is the required vector.