﻿ Establish the vector inequalities geometrically or otherwise :(d) |a−b| > ||a| − |b|| from Physics Motion in A Plane Class 11 Manipur Board

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Establish the vector inequalities geometrically or otherwise :
(d) |ab| > ||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.

That is,

| | 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 | |

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