2.1 Two charges 5 × 10–8 C and –3 × 10–8 C are located 16 cm apart. At what point(s) on the line joining the two charges is the electric potential zero? Take the potential at infinity to be zero.

4.4 A horizontal overhead power line carries a current of 90 A in east to west direction. What is the magnitude and direction of the magnetic field due to the current 1.5 m below the line?

2.7 kg, then its volume is 10–3 m3 (a scalar) and its density is 2.7×103 kg m–3 (a scalar). A vector quantity is a quantity that has both a magnitude and a direction and obeys the triangle law of addition or equivalently the Fig. 3.1 (a) Position and displacement vectors. parallelogram law of addition. So, a vector is (b) Displacement vector PQ and different specified by giving its magnitude by a number courses of motion. and its direction. Some physical quantities that It is important to note that displacement are represented by vectors are displacement, vector is the straight line joining the initial and velocity, acceleration and force. final positions and does not depend on the actual To represent a vector, we use a bold face type path undertaken by the object between the two in this book. Thus, a velocity vector can be positions. For example, in Fig. 3.1(b), given the represented by a symbol v. Since bold face is initial and final positions as P and Q, the difficult to produce, when written by hand, a displacement vector is the same PQ for different vector is often representedrv by an arrow placedrv paths of journey, say PABCQ, PDQ, and PBEFQ.over a letter, say . Thus, both v and Therefore, the magnitude of displacement is represent the velocity vector. The magnitude of either less or equal to the path length of an a vector is often called its absolute value, object between two points. This fact was indicated by |v| = v. Thus, a vector is emphasised in the previous chapter also whilerepresented by a bold face, e.g. by A, a, p, q, r, ... discussing motion along a straight line.x, y, with respective magnitudes denoted by light face A, a, p, q, r, ... x, y. 3.2.2 Equality of Vectors 3.2.1 Position and Displacement Vectors Two vectors A and B are said to be equal if, and only if, they have the same magnitude and theTo describe the position of an object moving in same direction.**a plane, we need to choose a convenient point, say O as origin. Let P and P′ be the positions of Figure 3.2(a) shows two equal vectors A and the object at time t and t′, respectively [Fig. 3.1(a)]. B. We can easily check their equality. Shift B We join O and P by a straight line. Then, OP is parallel to itself until its tail Q coincides with that the position vector of the object at time t. An of A, i.e. Q coincides with O. Then, since their arrow is marked at the head of this line. It is tips S and P also coincide, the two vectors are represented by a symbol r, i.e. OP = r. Point P′ is said to be equal. In general, equality is indicated * Addition and subtraction of scalars make sense only for quantities with same units. However, you can multiply and divide scalars of different units. ** In our study, vectors do not have fixed locations. So displacing a vector parallel to itself leaves the vector unchanged. Such vectors are called free vectors. However, in some physical applications, location or line of application of a vector is important. Such vectors are called localised vectors. Reprint 2025-26 MOTION IN A PLANE 29 The factor λ by which a vector A is multiplied could be a scalar having its own physical dimension. Then, the dimension of λ A is the product of the dimensions of λ and A. For example, if we multiply a constant velocity vector by duration (of time), we get a displacement vector. 3.4 ADDITION AND SUBTRACTION OF VECTORS — GRAPHICAL METHOD Fig. 3.2 (a) Two equal vectors A and B. (b) Two As mentioned in section 4.2, vectors, by vectors A′ and B′ are unequal though they definition, obey the triangle law or equivalently, are of the same length. the parallelogram law of addition. We shall now describe this law of addition using the graphical as A = B. Note that in Fig. 3.2(b), vectors A′ and method. Let us consider two vectors A and B that B′ have the same magnitude but they are not lie in a plane as shown in Fig. 3.4(a). The lengths equal because they have different directions. of the line segments representing these vectors Even if we shift B′ parallel to itself so that its tail are proportional to the magnitude of the vectors. Q′ coincides with the tail O′ of A′, the tip S′ of B′ To find the sum A + B, we place vector B so that does not coincide with the tip P′ of A′. its tail is at the head of the vector A, as in 3.3 MULTIPLICATION OF VECTORS BY REAL Fig. 3.4(b). Then, we join the tail of A to the head NUMBERS of B. This line OQ represents a vector R, that is, the sum of the vectors A and B. Since, in thisMultiplying a vector A with a positive number λ procedure of vector addition, vectors aregives a vector whose magnitude is changed by the factor λ but the direction is the same as that of A : λ A = λ A if λ > 0. For example, if A is multiplied by 2, the resultant vector 2A is in the same direction as A and has a magnitude twice of |A| as shown in Fig. 3.3(a). Multiplying a vector A by a negative number −λ gives another vector whose direction is opposite to the direction of A and whose magnitude is λ times |A|. Multiplying a given vector A by negative numbers, say –1 and –1.5, gives vectors as shown in Fig 3.3(b). (c) (d) Fig. 3.3 (a) Vector A and the resultant vector after multiplying A by a positive number 2. Fig. 3.4 (a) Vectors A and B. (b) Vectors A and B (b) Vector A and resultant vectors after added graphically. (c) Vectors B and A multiplying it by a negative number –1 added graphically. (d) Illustrating the and –1.5. associative law of vector addition. Reprint 2025-26 30 PHYSICS arranged head to tail, this graphical method is What is the physical meaning of a zero vector? called the head-to-tail method. The two vectors Consider the position and displacement vectors and their resultant form three sides of a triangle, in a plane as shown in Fig. 3.1(a). Now suppose so this method is also known as triangle method that an object which is at P at time t, moves to of vector addition. If we find the resultant of P′ and then comes back to P. Then, what is its B + A as in Fig. 3.4(c), the same vector R is displacement? Since the initial and final obtained. Thus, vector addition is commutative: positions coincide, the displacement is a “null vector”. A + B = B + A (3.1) Subtraction of vectors can be defined in termsThe addition of vectors also obeys the associative of addition of vectors. We define the differencelaw as illustrated in Fig. 3.4(d). The result of of two vectors A and B as the sum of two vectorsadding vectors A and B first and then adding A and –B :vector C is the same as the result of adding B and C first and then adding vector A : A – B = A + (–B) (3.5) (A + B) + C = A + (B + C) (3.2) It is shown in Fig 3.5. The vector –B is added to vector A to get R2 = (A – B). The vector R1 = A + BWhat is the result of adding two equal and is also shown in the same figure for comparison.opposite vectors ? Consider two vectors A and We can also use the parallelogram method to–A shown in Fig. 3.3(b). Their sum is A + (–A). find the sum of two vectors. Suppose we haveSince the magnitudes of the two vectors are the same, but the directions are opposite, the two vectors A and B. To add these vectors, we resultant vector has zero magnitude and is bring their tails to a common origin O as represented by 0 called a null vector or a zero shown in Fig. 3.6(a). Then we draw a line from vector : the head of A parallel to B and another line from the head of B parallel to A to complete a A – A = 0 |0|= 0 (3.3) parallelogram OQSP. Now we join the point of Since the magnitude of a null vector is zero, its the intersection of these two lines to the origin direction cannot be specified. O. The resultant vector R is directed from the The null vector also results when we multiply common origin O along the diagonal (OS) of the a vector A by the number zero. The main parallelogram [Fig. 3.6(b)]. In Fig.3.6(c), the properties of 0 are : triangle law is used to obtain the resultant of A A + 0 = A and B and we see that the two methods yield the λ 0 = 0 same result. Thus, the two methods are 0 A = 0 (3.4) equivalent. Fig. 3.5 (a) Two vectors A and B, – B is also shown. (b) Subtracting vector B from vector A – the result is R2. For comparison, addition of vectors A and B, i.e. R1 is also shown. Reprint 2025-26 MOTION IN A PLANE 31 Fig. 3.6 (a) Two vectors A and B with their tails brought to a common origin. (b) The sum A + B obtained using the parallelogram method. (c) The parallelogram method of vector addition is equivalent to the triangle method. ⊳ Example 3.1 Rain is falling vertically with 3.5 RESOLUTION OF VECTORS a speed of 35 m s–1. Winds starts blowing Let a and b be any two non-zero vectors in a after sometime with a speed of 12 m s–1 in plane with different directions and let A be east to west direction. In which direction another vector in the same plane (Fig. 3.8). A should a boy waiting at a bus stop hold can be expressed as a sum of two vectors — one his umbrella ? obtained by multiplying a by a real number and the other obtained by multiplying b by another real number. To see this, let O and P be the tail and head of the vector A. Then, through O, draw a straight line parallel to a, and through P, a straight line parallel to b. Let them intersect at Q. Then, we have A = OP = OQ + QP (3.6) But since OQ is parallel to a, and QP is parallel to b, we can write : OQ = λ a, and QP = µ b (3.7) Fig. 3.7 where λ and µ are real numbers. Answer The velocity of the rain and the wind Therefore, A = λ a + µ b (3.8)are represented by the vectors vr and vw in Fig.