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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 PORS, as shown in the given figure.

$|\vec{\text{OR}}|=|\vec{\text{PS}}|=|\vec{\text{b}}|\ ...(\text{i})$

$|\vec{\text{OP}}|=|\vec{\text{a}}|\ ...(\text{ii})$

In a triangle, each side is smaller than the sum of the other two sides.

Therefore, in $\triangle\text{OPS},$

$\text{OS}<\text{OP}+\text{PS}$

$|\vec{\text{a}}-\vec{\text{b}}|<|\vec{\text{a}}|+|-\vec{\text{b}}|$

$|\vec{\text{a}}-\vec{\text{b}}|<|\vec{\text{a}}|+|\vec{\text{b}}|\ ...(\text{iii})$

If the two vectors act in a straight line but in opposite directions, then we can write:

$|\vec{\text{a}}-\vec{\text{b}}|=|\vec{\text{a}}|+|\vec{\text{b}}|\ ...(\text{iv})$

Combining equations (iii) and (iv), we get:

$|\vec{\text{a}}-\vec{\text{b}}|\le|\vec{\text{a}}|+|\vec{\text{b}}|$

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