4.5 Vectors in Cartesian Plane

(A) Vectors in Cartesian Coordinates

1. A unit vector is a vector whose magnitude is one unit.
2. A unit vector that is parallel to the x-axis is denoted by   i ˜ while a unit 
    vector that is parallel to the y-axis is denoted by   j ˜ .
3. The unit vector can be expressed in columnar form as below:
OA =x i ˜ +y j ˜ =( x y )            (Column Vector)   
4. The magnitudes of the unit vectors are  | i ˜ |=| j ˜ |=1.

5. The magnitude of the vector  OA can be calculated using the
    Pythagoras’ Theorem.
  | OA |= x 2 + y 2   

(B) Unit Vector in the Direction of a Vector
  Unit vector of  a ˜ ,  a ^ ˜ = x i ˜ +y j ˜ x 2 + y 2   
Example 1:
If    r ˜ =k i ˜ 8 j ˜ and | r ˜ |=10 , find the values of k. Determine the unit vector in the direction of   r ˜    for each value of k.          

Solution:
Given | r ˜ |=10 x 2 + y 2 =10 k 2 + ( 8 ) 2 =10 k 2 +64=100 k=±6 Unit vector of   r ˜ ^ = x i ˜ +y j ˜ x 2 + y 2 When k=6,                                When k=6 r ˜ ^ = 6 i ˜ 8 j ˜ 10 = 3 i ˜ 4 j ˜ 5                 r ˜ ^ = 6 i ˜ 8 j ˜ 10 = 3 i ˜ 4 j ˜ 5 r ˜ ^ = 1 5 ( 3 i ˜ 4 j ˜ )                           r ˜ ^ = 1 5 ( 3 i ˜ 4 j ˜ )

Example 2:
It is given that a ˜ =( 6 3 ) and  b ˜ =( 3 7 ).  
(a) Find b ˜ a ˜  and | b ˜ a ˜ |.  
(b) Hence, find the unit vector in the direction of b ˜ a ˜  .

Solution:
(a)
b ˜ a ˜ =( 3 7 )( 6 3 )=( 36 73 )=( 3  4 ) | b ˜ a ˜ |= ( 3 ) 2 + 4 2 = 9+16 = 25 =5

(b)
The unit vector in the direction of  b ˜ a ˜ = 1 5 ( 3  4 ) =( 3 5   4 5 )