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VIII - PHYSICS (Vol-1) Olympiad Text Book Narayana Group of Schools 25 VECTORS_SYNOPSIS-3 1. ADDITION OF VECTORS Scalars can be added or subtracted following the simple rules of algebra or arithmetic. But vectors do not follow the same simple rules, because while adding or subtracting vectors, their direction also has to be considered. For example, when a mass of 5 kg is added to another mass of 5 kg, the result is exactly 10kg. But when a vector of magnitude 5 units is added to another vector of magnitude 5 units (of course of same physical quantity), the resultant may have a magnitude from zero to a maximum of 10 units, depending on relative orientations of the two vectors. The vector sum is also called resultant. 2. ADDITION OF TWO VECTORS IN SAME DIRECTION If two vectors are in the same direction, their resultant (sum) is obtained by adding their vector lengths as shown in the figure (a). The direction of resultant is same as the individual vectors. Fig (a)- Addition of vectors in same direction A  B  A  B  C A B      A  B  A  B  C A B      figure (a) 3. ADDITION OF TWO VECTORS IN OPPOSITE DIRECTION If the vectors are mutually opposite, their resultant is obtained by subtracting the length of smaller vector from that of larger vector as shown in figure (b). The direction of resultant is same as that of larger vector. (Triangle law) A  B  C A ( B)       figure (b) 4. ADDITION OF TWO VECTORS INCLINED MUTUALLY If two vectors are mutually inclined, the following procedure is adopted to find their sum. A and B are the given vectors. B is slides parallel to itself, such that its ‘tail’ coincides with the head of A as shown in figure (c). Then the directed line segment drawn from the tail of A to the head of B represents the addition of A and B. A  B  C A B      A  B 
26 Narayana Group of Schools VIII - PHYSICS (Vol-1) Olympiad Text Book 5. Triangle law of vectors: If two vectors are represented in magnitude and direc- tion by the two sides of a triangle taken in order, the third side of the triangle taken in reverse order represents their resultant in magnitude and direction.(or) If three vectors simultaneously acting at a point have zero resultant then these three vectors can be represented both in magnitude and direction by the sides of a triangle taken in an order. A C B In the above figure A B C O        6. PARALLELOGRAM LAW OF VECTORS : Two vector quantities can be added using parallelogram law ( velocity vector can be added to velocity vector only). This law is useful to find both magnitude and direction of resultant. Statement: If two vectors are represented in magnitude and direction by the adjacent sides of a parallelogram drawn from a point, the diagonal passing through that point represents their resultant both in magnitude and direction. A  B D Q  P  A  B D Q  P  A  B D Q  P  C E R   figure (d) Explanation: P  and Q  are two vectors represented by AB  and AD  . Both vectors act at the common point A and mutually inclined at angle ‘  ’ as shown in fig (d). If the parallelogram ABCD is completed taking AB and AD as adjacent sides, then the diagonal AC  represents their resultant R  both in magnitude and direction. Magnitude of the resultant: The line of action of P  is extended. The perpendicular drawn from ‘C’ meets the extension of AB at E. From the figure, it is obvious that BC AD Q      and    CBE  Length of AB = magnitude of P  = P ; Length of BC = magnitude of Q  = Q Length of AC = magnitude of R  = R From triangle CBE, cos  = BE BC , BE = BC cos   BE = Q cos  ........... [1] From the triangle AEC, EC sin BC   and EC = Q sin ........... [2] (AC)2 = (AE)2 + (EC)2  (AC)2 = (AB + BE)2 + (EC)2  (AC)2 = (AB)2 + (BE)2 + 2AB . BE + (EC)2  R2 = P2 + Q2 cos2  + 2PQ cos  + Q2 sin2  R2 = P2 + Q2 + 2PQ cos   R 2 2      P Q 2PQcos (3)

28 Narayana Group of Schools VIII - PHYSICS (Vol-1) Olympiad Text Book VECTORS WORKSHEET-3 :1. If A+B = A + B then the angle between the A and B . 1) 0 0 2) 900 3) 1800 4) 600 2. p  and q  are two adjacent sides of a parallelogram. if p q    is one diagonal, then the other diagonal is 1) p q    2) q p    3) q p    4) 1 & 2 3. a-b = a + b then the angle between a and b is 1) 900 2) 1800 3) 450 4) 00 4. If P + Q = R and P - Q = S then 2 2 R + S is equal to 1) 2 2 P + Q 2) 2 2 2(P - Q ) 3) 2 2 2(P + Q ) 4) 4PQ 5. If the magnitudes of _ A and _ B are a and b respectively, the magnitude of the resultant vector is always 1) equal to a b   2) less than a b   3) greater than a b   4)not greater than a b   6. _ _ _ _ A B A B    then the angle between _ _ A B, is 1) 1200 2)00 3)900 4)1800 7. Out of the following the resultant of which cannot be 4N. 1) 2N and 2N 2) 3N and 8N 3) 2N and 6N 4) 4N and 8N 8. The maximum value of magnitude of A B   is 1) A B 2) A 3) A B  4) 2 2 A B  9. Two vectors A and B have precisely equal magitudes. If magnitude of A B  to be larger than the magnitude of A B by a factor n, the angle between them is 1)   1 2 tan 1 n  2)   1 tan 1 n  3)   1 tan 1 2n  4)   1 2 tan 1 2n  10. If the Resultant of P  and Q  makes an angle 1 with P  and 2 with Q  . Then 1. 1 2     if P Q 2. 1 2     if P Q 3. 1 2     if P Q 4. 1 2     if P Q 11. It is found that A B A   . This necessarily implies. 1. B  0 2. A, B are antiparallel 3. A, B are perpendicular 4. A B. 0 