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2(b).19. POSITION VECTOR, Position vector of any point, with respect to an, arbitrarily chosen origin, is defined as the vector which, connects the origin and the point and is directed towarde, the point., Position vector of a point, helps in locating the position, of the point. Its magnitude, gives the distance between, origin and the point. Consider, a point P' having co-ordinates, (x, y, z) [Fig. 2(b).38]. If 'O' is, P(x y, the origin, then OP which is, called the position vector r, will have x, y and z as its, three orthogonal components., Fig. 2(b).38. Position vector., x +y + z, or, zk, Magnitude of r is given by, e, 17%=Vx + y + z., Scanned by TapScanner
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TOOYS, (6).23. DOT PRODUCT OR SCALAR, PRODUCT, Dot product of two vectors is defined as the product of, their magnitudes and the cosine of the smaller angle, between the two., It is written by putting a dot (•) between two vectors., The result of this product does, not possess any direction. So, it, is a scalar quantity. Hence it is, also called a scalar product., K, Consider two vectors A, (= Oa) and B (= Ob) drawn, from a point O and inclined to, each other at an angle A (Fig Fig. 2(b).41. Dot product of, 2(b).41]., %3D, L, A, two vectors., Dot product of A and B is given by, A.B IATIBI cos 0 AB cos 0, ...(11), Scanned by TapScanner
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Physical quantities represented by dot, O Work. Let a body be displaced from P to Q, be represented by the dot product of two vectors., product of two vectors, There are so many quantities in Physics which, Following few examples shall illustrate it., (displacement S) under the, action of a force F [Fig., 2(b).42]. Work done W is, given by, W = (F cos 0) S, Q, %3D, W = FS cos 0, Fig. 2(b).42. Work done, or W = F. S., So, work can be represented as the dot product of, force (F) and displacement (S). Therefore, work is a, scalar quantity., %D, (ii) Power. Power is defined as rate of doing work., work F.S, F (S), Power, time, Power = F.U, bas, or, %3D, where v represents velocity. So power is also a scalar, quantity., (iii) Magnetic flux. Magnetic flux oB associated, with an area dS is, d oB = B dS cos 0, %3D, where 0 is the angle between magnetic induction, B and, vector dS representing area., d OB = B. dS, %3D, So, magnetic flux is a scalar quantity., (iv) Potential energy of a dipole (electric/, magnetic). When an electric dipole of dipole moment p, is situated in an electric field E or a magnetic dipole of, magnetic moment M in a magnetic field B, the potential, energy (a scalar quantity) associated with it is given, by, Ug = -p. E (in case if electric dipole), %3D, UB = – M. B (in case of magnetic dipole), Characteristies of dot product : bes, Dot product of two vectors obeys the following, characteristics., Scanned by TapScanner, %3D, P.
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Commutatice, Dot product between two vectors is commutative in, nature., If A and B are two vectors,, A.B = B. A, i.e., It means that the order of vectors in the product can, he changed without affecting the result., ( Distributive, Dot product of a vector with the sum of a number of, other vectors is equal to the sum of the dot products of, the vector taken with other vectors separately., Mathematically, it can be expressed as, A. (B+ C +...) = Ả.B+A.C+..., (iji Perpendicular vectors, For two perpendicular vectors A and B, 0 = 90°., A. B = AB cos 0 = AB cos 90° = AB.0 = 0, Thus, the dot product of two non-zero vectors, which, are perpendicular to each other, is always zero. This, statement is known as condition of perpendicularity., Since î,j and k are mutually perpendicular,, 全.3=9.=2.会=0, (i Collinear vectors, (a) Parallel vectors, In this case, 0 = 0°, %3D, A.B= AB cos 0 = AB cos 0° = AB, (b) Anti-parallel vectors, In this case,, 0 = 180°, A.B = AB cos 180° = -AB, Therefore, the dot product of collinear vectors is, equal to the product of their magnitudes. It is positive if, they are parallel and negative if they are anti-parallel., The statement is called condition of collinearity., KEqual vectors, Vectors are equal if they possess same magnitude, and direction, i.e.,, 0 = 0°, %3D, . Dot product of two equal vectors is given by, A . A = AA cos 0° = A², %3D, %3D, Dot product of two equal vectors is equal to the, square of the magnitude of the either., Scanned by TapScanner, る)
