By Comite d'Organization du Congres

Papers from the foreign arithmetic convention in great in 1970

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K. 2 The Concept of a Function We now investigate the exact mathematical description of the dependence of two quantities. Example Consider a spring fixed at one end and stretched at the other end, as shown in Fig. 1. This results in a force which opposes the stretching or displacement. Two quantities can be measured: the displacement x in metres (m); the force F in newtons (N). Measurements are carried out for several values of x. Thus we obtain a series of paired values for x and F associated with each other.

If the position vector of a point P is r = (0, ry , rz ) and the angular velocity ! z ), as shown in the figure, then the velocity v of P is i j k v = ! z i 0 ry rz Fig. 1 Scalar Product 1. 5 b=3 ˛ = 120◦ 2. Considering the scalar products, what can you say about the angle between the vectors a and b? (a) a · b = 0 (b) a · b = ab ab (c) a · b = (d) a · b < 0 2 3. Calculate the scalar product of the following vectors: (a) a = (3, −1, 4) (b) a = (3/2, 1/4, −1/3) b = (−1, 2, 5) b = (1/6, −2, 3) (c) a = (−1/4, 2, −1) (d) a = (1, −6, 1) b = (1, 1/2, 5/3) b = (−1, −1, −1) 2.

Fig. 26 Fig. 27 4. Draw the vector c = a − b. Fig. 28 Fig. 4 Components and Projections of a Vector 5. Project vector a on to vector b. Fig. 30 Fig. 31 Exercises 19 6. Calculate the magnitude of the projection of a on to b. 6 Component Representation 7. Given the points P1 = (2, 1), P2 = (7, 3) and P3 = (5, −4), calculate the coordinates of the fourth corner P4 of the parallelogram P1 P2 P3 P4 formed by −−→ −−→ the vectors a = P1 P2 and b = P1 P3 . Fig. 32 8. If P1 = (x1 , y1 ), P2 = (x2 , y2 ), P3 = (x3 , y3 ) and P4 = (x4 , y4 ) are −−→ −−→ −−→ four arbitrary points in the x-y plane and if a = P1 P2 , b = P2 P3 , c = P3 P4 , −−→ d = P4 P1 , calculate the components of the resultant vector S = a + b + c + d and show that S = 0.