Question
Establish the vector inequalities geometrically or otherwise :
(b) |a+b| > ||a| −|b||
Solution
Let two vectors a and b be represented by the adjacent sides of a parallelogram OMNP, as shown in the given figure. 
Here, we have:
| OM | = | a |
| MN | = | OP | = | b |
| ON | = | a + b |
In a triangle, each side is smaller than the sum of the other two sides.
Therefore, in ΔOMN, we have,
ON + MN > OM
ON + OM > MN
| ON | > | OM - OP | (∵ OP = 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 | |
