Question
Establish the vector inequalities geometrically or otherwise :
(c) |a−b| < |a| + |b|
Solution
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 |
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 |
