﻿ Establish the vector inequalities geometrically or otherwise : (b) |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 :

(b) |a+b| > ||a| |b||

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 | |

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