Question
Establish the following vector inequalities geometrically or otherwise: $|\text{a}-\text{b}|\ge||\text{a}|-|\text{b}||$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. 
The following relations can be written for the given parallelogram. (OS + PS) > OP .....(i) OS < (OP - PS) .....(ii) $|\vec{\text{a}}-\vec{\text{b}}|>|\vec{\text{a}}|-|\vec{\text{b}}|\ ...(\text{iii})$ 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 as: $||\vec{\text{a}}-\vec{\text{b}}||>||\vec{\text{a}}|-|\vec{\text{b}}||$ $|\vec{\text{a}}-\vec{\text{b}}|>||\vec{\text{a}}|-|\vec{\text{b}}||\ ...(\text{iv})$ If the two vectors act in a straight line but in opposite directions, then we can write: $|\vec{\text{a}}-\vec{\text{b}}|=||\vec{\text{a}}|-|\vec{\text{b}}||\ ...(\text{v})$ Combining equations (iv) and (v), we get: $|\vec{\text{a}}-\vec{\text{b}}|\ge||\vec{\text{a}}|-|\vec{\text{b}}||$
The following relations can be written for the given parallelogram. (OS + PS) > OP .....(i) OS < (OP - PS) .....(ii) $|\vec{\text{a}}-\vec{\text{b}}|>|\vec{\text{a}}|-|\vec{\text{b}}|\ ...(\text{iii})$ 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 as: $||\vec{\text{a}}-\vec{\text{b}}||>||\vec{\text{a}}|-|\vec{\text{b}}||$ $|\vec{\text{a}}-\vec{\text{b}}|>||\vec{\text{a}}|-|\vec{\text{b}}||\ ...(\text{iv})$ If the two vectors act in a straight line but in opposite directions, then we can write: $|\vec{\text{a}}-\vec{\text{b}}|=||\vec{\text{a}}|-|\vec{\text{b}}||\ ...(\text{v})$ Combining equations (iv) and (v), we get: $|\vec{\text{a}}-\vec{\text{b}}|\ge||\vec{\text{a}}|-|\vec{\text{b}}||$Need a full question paper?
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