Find a vector of magnitude 5 units, and parallel to the resultant of the vectors .
It is given that a → = 2 i ^ + 3 j ^ − k ^ and b → = i ^ − 2 j ^ + k ^ .
The resultant of vector a → and b → is a → + b → .
( a → + b → ) = ( 2 + 1 ) i ^ + ( 3 − 2 ) j ^ + ( − 1 + 1 ) k ^ ( a → + b → ) = 3 i ^ + 1 j ^ + 0 k ^
Consider c → = ( a → + b → ) , then c → = 3 i ^ + 1 j ^ + 0 k ^ .
The magnitude of c → is,
| c → | = 3 2 + 1 2 + 0 2 | c → | = 9 + 1 | c → | = 10
The unit vector in direction of c → is,
c ^ = 1 | c → | × c → c ^ = 1 10 × [ 3 i ^ + j ^ + 0 k ^ ] c ^ = 3 10 i ^ + 1 10 j ^
If vector with unit magnitude is c ^ , then vector with 5 magnitude is,
5 c ^ = 5 × ( 3 10 i ^ + 1 10 j ^ ) 5 c ^ = 15 10 i ^ + 5 10 j ^ 5 c ^ = 3 2 10 i ^ + 10 2 j ^
Thus, the vector parallel to the resultant vector with magnitude 5 units is 3 2 10 i ^ + 10 2 j ^ .
It is given that
The resultant of vector
Consider
The magnitude of
The unit vector in direction of
If vector with unit magnitude is
Thus, the vector parallel to the resultant vector with magnitude
.