Establish the vector inequality:

(a) |a + b| ≤ |a| + |b|

Let two vectors a and b be represented by the adjacent sides of a parallelogram OMNP, as shown in the given figure.



Here, we can write: 
OM = | a |                                 ...(i) 
    
MN = OP = | b |                         ...(ii) 
ON = | a + b |                            ...(iii) 
In a triangle, each side is smaller than the sum of the other two sides. 
Therefore, in ΔOMN, we have,
ON < (OM + MN) 
| a + b | < | a | + | b |                 ...(iv) 
If the two vectors a and b act along a straight line in the same direction, then we can write: 
| a + b | = | a | + | b |                  ... (v) 
Combining equations (iv) and (v), we get: 
| a + b | ≤ | a | + | b |