Q. 63.7( 45 Votes )

Establish the following vector inequalities geometrically or otherwise:

A.

B.

C.

D.

When does the equality sign above apply?

Answer :

Let us consider two vectors and such that = and =. Also,=. According to Parallelogram law of vector addition, = and = as shown in the figure.



From the figure,


OA=


OB=AC=


OC=


OC’=


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


So, in ΔAOC,


OC < OA + AC



If both the vectors act along a straight line, then the equality condition occurs as


So, =



B. In ΔAOC,


OC+AC>OA


|OC|>|OA-AC|


|OC|>|OA-OB| ( AC=OB are the parallel sides of the parallelogram)



If both the vectors act along a straight line, then the equality condition occurs as




C. In ΔOAC’,


OC’<OA+AC’



( )


If both the vectors act along a straight line, then the equality condition occurs as




D. In ΔOAC’,


OC’+AC’>OA


|OC’|>|OA-AC’|




If both the vectors act along a straight line, then the equality condition occurs as




NOTE: Parallelogram Law of Vector Addition states that when two vectors are represented by two adjacent sides of a parallelogram by direction and magnitude then the resultant of these vectors is represented in magnitude and direction by the diagonal of the parallelogram starting from the same point.


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