Establish the vector inequalities geometrically or otherwise :

(c) |a−b| < |a| + |b|

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

 
Here we have: 
| OR | = | PS | = | b |              ...(i) 
| OP | = | a |                          ...(ii) 
In a triangle, each side is smaller than the sum of the other two sides. Therefore, in ΔOPS, we have: 
OS < OP + PS 
| a - b | < | a | + | -b | 
| a - b | < | a | + | b |            ... (iii) 
If the two vectors act in a straight line but in opposite directions, then we have,
| a - b | = | a | + | b |             ... (iv) 
Combining equations (iii) and (iv), we get, 
| a - b | ≤ | a | + | b |