Establish the vector inequalities geometrically or otherwise :
(d) |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.
The following relations can be written for the given parallelogram.
OS + PS > OP ...(i)
OS > OP - PS ...(ii)
| a - b | > | a | - | b | ...(iii)
The quantity on the LHS is always positive and that on the RHS can be positive or negative.
To make both quantities positive, we take modulus on both sides.
| | a - b | | > | | a | - | b | |
| a - b | > | | a | - | b | | ...(iv)
If the two vectors act in a straight line but in the opposite directions, then we can write,
| a - b | = | | a | - | b | | ...(v)
Combining equations (iv
) and (v
), we get,
| a - b | ≥ | | a | - | b | |