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Chapter 1 The algebra and calculus of vectors 12 Problem 1. 5 In a general tetrahedron, lines are drawn connecting the mid-point of each side with the mid-point of the side opposite. Show that these three lines meet in a point that bisects each of them. These three points are therefore coincident. Hence the three lines that join the mid-points of opposite sides of the tetrahedron meet in a point that bisects each of them.Chapter 1 The algebra and calculus of vectors 13 Problem 1. 6 Let ABCD be a general tetrahedron and let P, Q, R, S be the median centres of the faces opposite to the vertices A, B, C, D respectively. These four points are therefore coincident. Hence the four lines that join the vertices of the tetrahedron to the median centres of the opposite faces meet in a point that divides each line in the ratio 3:1. It is the same point as was constructed in Problem 1.5. cCambridge University Press, 2006 14. Chapter 1 The algebra and calculus of vectors 14 Problem 1. 7 A number of particles with masses m1; m2; m3;::: are situated at the points with position vectors r1; r2; r3;::: relative to an origin O. The centre of mass G of the particles is dened to be the point of space with position vector R D m1r1 C m2r2 C m3r3 Cm1 C m2 C m3 CShow that if a different origin O0 were used, this denition would still place G at the same point of space. Solution Suppose the line segment.Chapter 1 The algebra and calculus of vectors 15 Problem 1. 8 Prove that the three perpendiculars of a triangle are concurrent. FIGURE 1.3 AL and BM are two of the perpendiculars of the triangle ABC. A B C O L M N Solution Let ABC be the triangle and construct the perpendiculars AL and BM from A and B; let O be their point of intersection. Now construct the line CO and extend it to meet AB in the point N. We wish to show that CN is perpendicular to AB.

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To browse Academia.edu and the wider internet faster and more securely, please take a few seconds to upgrade your browser. You can download the paper by clicking the button above. Related Papers C L A S S I C A L M E C H A N I C S SOLUTIONS MANUAL By Makoto Sato Undergraduate Lecture Notes in Physics Mechanics and Thermodynamics By Sofia Bozadjie Using simulation for planning and design of robotic systems with intermittent contact By Stephen Berard Nonconservative stability problems of modern physics By Oleg Kirillov Walter Fox Smith Waves and oscillations A preluBookZZ.org By Flavio Cruz READ PAPER Download pdf. E A C D O B a Solution Figure 1.1 shows a cube of side a; OE and AD are two of its diagonals. Let O be the origin of position vectors and suppose the points A, B and C have position vectors ai, aj, ak respectively. Then the line segment. OE represents the vector ai C aj C ak and the line segment.Find the position vectors of the vertices C, D, E, F in terms of the position vectors a, b of A and B. FIGURE 1.2 ABCDEF is a regular hexagon. A BC D E F O a b Solution (i) The position vector c is represented by the line segment. OC which has the same magnitude and direction as the line segment ! AB. Hence c D b a: (ii) The position vector d is represented by the line segment. OD which has the same magnitude as, but opposite direction to, the line segment ! OA. Hence d D a: (iii) The position vector e is represented by the line segment. OE which has the same magnitude as, but opposite direction to, the line segment ! OB. Hence e D b: (iv) The position vector f is represented by the line segment. OF which has the cCambridge University Press, 2006 10. Chapter 1 The algebra and calculus of vectors 11 Problem 1. 4 Let ABCD be a general (skew) quadrilateral and let P, Q, R, S be the mid-points of the sides AB, BC, CD, DA respectively. Show that PQRS is a parallelogram. Similarly, the lines QR and PS are parallel. The quadrilateral PQRS is therefore a parallelogram.

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