4.3.2 Subtraction of Vectors


4.3.2 Subtraction of Vectors
The subtraction of the vector b ˜ from the vector  a ˜ is written as  a ˜ b ˜ . This operation can be considered as the addition of the vector a ˜ with the negative vector of  b ˜. Therefore a ˜ b ˜ = a ˜ +( b ˜ ).  

Example 1:

In the diagram above, vector OP = p ˜ OR = r ˜ and Q divides PR in the ratio of 2: 3. Find the following vectors in terms of p ˜  and  r ˜
( a )  PR ( b )  OQ ( c )  QM  if M is the midpoint of OR.

Solution:
(a)
PR = PO + OR      = OP + OR      = p ˜ + r ˜

(b)
OQ = OP + PQ      = OP + 2 5 PR      = p ˜ + 2 5 ( p ˜ + r ˜ )      = p ˜ 2 5 p ˜ + 2 5 r ˜      = 3 5 p ˜ + 2 5 r ˜

(c)
QM = QO + OM      = OQ + OM      = OQ + 1 2 OR      =( 3 5 p ˜ + 2 5 r ˜ )+ 1 2 r ˜      = 3 5 p ˜ 2 5 r ˜ + 1 2 r ˜      = 3 5 p ˜ + 1 10 r ˜