Q6 of 32 Page 85

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.


More from this chapter

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4

State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful:

(a) adding any two scalars, (b) adding a scalar to a vector of the same dimensions,


(c) multiplying any vector by any scalar, (d) multiplying any two scalars, (e) adding any two vectors, (f) adding a component of a vector to the same vector.

 5

Read each statement below carefully and state with reasons, if it is true or false:

(a) The magnitude of a vector is always a scalar, (b) each component of a vector is always a scalar, (c) the total path length is always equal to the magnitude of the displacement vector of a particle. (d) the average speed of a particle (defined as total the path) is either greater or equal to the magnitude of average velocity of the particle over the same interval of time, (e) Three vectors not lying in a plane can never add up to give a null vector.

 7

Given a + b + c + d = 0, which of the following statements are correct:

A. a, b, c, and d must each be a null vector,


B. The magnitude of (a + c) equals the magnitude of (b + d),


C. The magnitude of a can never be greater than the sum of the magnitudes of b, c, and d,


D. b + c must lie in the plane of a and d if a and d are not collinear, and in the line of a and d, if they are collinear?

 8

Three girls skating on a circular ice ground of radius 200 m start from a point P on the edge of the ground and reach a point Q diametrically opposite to P following different paths as shown in Fig. 4.20. What is the magnitude of the displacement vector for each? For which girl is this equal to the actual length of path skate ?