निम्नलिखित असमिकाओं की ज्यामिति या किसी अन्य विधि द्वारा स्थापना कीजिए :
(a) `|a+b|le|a|+|b|`
(b) `|a+b|ge||a|-|b||`
(c) `|A-b|le|a|+|b|`
(d) `|A-b|ge||a|-|b||`
इनमें समीका ( समता ) का चिह्न कब लागू होता है ?

Explanation

Let us consider that two vectors `vecV and vecB` are represented by two sides of a triangle : `OP=|vecA| and OQ=|vecB|`

(a) To prove `|vecA+vecB| le |vecA|+|vecB|`
As the length of one side of a triangle is laways less than the sum of length of two other sides .
Therefore `OR lt OP +PR`
`rArr OR lt OP+OQ`
`therefore |vecA+vecB|lt|vecA|+|vecB|`
If `veA and vecB` are acting along same straight lines ,
then `|vecA+vecB|=|vecA|+|vecB|`
Thus, we can say `|vecA+vecB|le|vecA|+|vecB|`
(b) To prove `|vecA+vecB| ge |vecA|-|vecB|`
`or OR+PR gt OP`
`OR gt |OP-OQ|`
`|vecA+vecB| gt |vecA|-|vecB|`
IF `vecA and vecB` are acting along a traight line but along opposite directions,
then `|vecA+vecB|=|vecA|-|vecB|`
Combinig (i) and (ii) we get
`|vecA+vecB| ge|vecA-|vecB|`
(c) To prove `|vecA-vecB| le | vecA|+|vecB|`
In `Delta OPR., OR lt OP +PR`.
`|vecA-vecB| lt |vecA|+|-vecB|`
`rArr |vecA-vecB| lt|vecA|+|vecB|`
When `vecA and vecB` are acting in a straight line but in opposite directions, then
`|vecA-vecB|=|vecA|+|vecB|`
From (i) and (ii) we can write `|vecA-vecB|lt |vecA|+|vecB|`
(d) To prove `|vecA-vecB| ge |vecA|-|vecB|` In OPR.
`OR. +PR. gt OP`
`rArr OR. gt |OP-PR.|`
`rArr OR. gt |OP-OS|`
`rArr |vecA-vecB|gt|vecA|-|vecB|`
If `vec A and vecB` are acting a straight line but in opposite directions , then
`|vecA-vecB|=||vecA|-|vecB|`
From (i) and (ii)
`therefore |vecA-vecB| ge||vecA|-|vecB|`