Complex Analysis III by Berenstein C.A. (ed)

By Berenstein C.A. (ed)

Lecture notes in arithmetic No.1277

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For what particular cases is it true that |a + b| = a + bl 16. Prove that |a—b| ^ a+b. 6 SCALAR PRODUCTS 31 17. If û, Φ are unit vectors with different directions, show that the vector ΰ+Φ bisects internally the angle between û and Φ. Is ^(û+Φ) a unit vector ? 5 The unit vectors i, j , k Let i, j , k denote unit vectors in the directions of the #-axis, jy-axis, 0-axis respectively. Then, i = (1,0,0), j = (0,1,0), k = (0,0,1). Using the rules for multiplication of a vector by a scalar and for addition of vectors, a vector SL = (aua2,a3) may be written a = aii + a2) + a3L·.

The angle 0 is always chosen in the range 0 ^ 0 ^ π. In each diagram, c is perpendicular to both a and b If 0 < α < 7Γ, sina > 0, and so a x b is in the direction Oz ; and if π < a < 2π> s i n a < 0 , so that a x b is in the opposite direction to Oz. However, if the angle between a and b is defined suitably, the geometrical interpretation may be expressed more conveniently, as follows. Let 0 be the angle between a and b, measured in the sense turning from a to b and chosen so that Ο^θ^π. Then the vector product a x b is the vector ab sin 0 c, where c is a unit vector at right angles to both a and b and such that to an observer looking in the direction of c, the sense in which Θ is measured is clockwise.

54. As in Exercise 53 above, the moment G of a force F about a point O is given by the equation G = r x F, where r is the position vector relative to O of any point on jSf the line of action of F. By direct substitution (or otherwise) show that the position vectors of points on ££ are given by r = XF + (FxG)IF2, where λ is a parameter. What is the perpendicular distance of O from *£ ? CHAPTER 3 VECTOR F U N C T I O N S OF A REAL VARIABLE. 1 Vector functions and their geometrical representation The reader should already be familiar with the idea of a real function f(x)> say, of a real variable x.

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