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.