# Establish the following vector inequalities geometrically or otherwise:A. B. C. D. When does the equality sign above apply?

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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