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.

Rate this question :

If |A|= 2 and |B| = 4, then match the relations in column I with the angle θ between A and B in column II

If |A| = 2 and |B| = 4, then match the relations in column I with the angle θ between A B and in column II.

A, B and C are three non-collinear, non-co-planar vectors. What can you say about direction of A × (B × C)?

Physics - ExemplarThree vectors A, B and C add up to zero. Find which is false.

Physics - Exemplar