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Motion in a Plane — Physics STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 SciencePhysicsMotion in a Plane5 Marks
Question
Establish the following vector inequalities geometrically or otherwise:
$|\text{a}+\text{b}|\le|\text{a}|+|\text{b}|$
When does the equality sign above apply?
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Answer

Let two vectors $\vec{\text{a}}$ and $\vec{\text{b}}$ be represented by the adjacent sides of a parallelogram OMNP, as shown in the figure.

$|\vec{\text{OM}}|=|\vec{\text{a}}|\ ...(\text{i})$
$|\vec{\text{MN}}|=|\vec{\text{OP}}|=|\vec{\text{b}}|\ ...(\text{ii})$
$|\vec{\text{ON}}|=|\vec{\text{a}}+\vec{\text{b}}|\ ...(\text{iii})$
In a triangle, each side is smaller than the sum of the other two sides.
Therefore, in $\triangle\text{OMN},$ we have:
$\text{ON}<(\text{OM}+\text{MN})$
$|\vec{\text{a}}+\vec{\text{b}}|<|\vec{\text{a}}|+|\vec{\text{b}}|\ ...(\text{iv})$
If the two vectors $\vec{\text{a}}$ and $\vec{\text{b}}$ act along a straight line in the same direction, then we can write:
$|\vec{\text{a}}+\vec{\text{b}}|=|\vec{\text{a}}|+|\vec{\text{b}}|\ ...(\text{v})$
Combining equations (iv) and (v), we get:
$|\vec{\text{a}}+\vec{\text{b}}|\le|\vec{\text{a}}|+|\vec{\text{b}}|$

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