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 O A can be calculated using the Pythagoras’ Theorem.
| O A | = 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 ) = ( 3 6 7 3 ) = ( 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 )

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