By C. W. Celia, A. T. F. Nice, K. F. Elliott
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Additional info for Advanced mathematics 2
9 Find in parametric form the cartesian equations of the straight line through the points (2, 1, 3) and (4, 7, -1). Show that this line passes through the points (5, 10, - 3) and (0, -5, 7). 10 Show that the two lines given by the equations 8 X x = 3 + 't, y = 4 + 2t, = 8 - 2s, y = 5 - s, Z z = - 2t, = -4 + 2s intersect. Find the coordinates of their common point. Find also the acute angle between these lines. 2 The vector equation of a plane Consider the plane which is perpendicular to the vector a and which passes through the point B with position vector b.
Find the cartesian equation of the plane which contains both lines. 34 Find the shortest distance to the line X 35 = 3- 2t, y = 1+ t, Z =4+ 2t (a) from the origin (b) from the point (1, 2, 3). Find a vector equation for the line of intersection of the planes with vector equations r·(2i + 2j + k) = 1, r·(4i + 7j + 4k) = 1. 1 Uniformly accelerated motion in a straight line Consider a particle moving along a straight line BOA (Fig. 1). The position of the particle is given by its distance from 0, distances to the right of 0 being taken to be positive and distances to the left of 0 being taken to be negative.
It is believed that x and y are connected by a relationship of the form x(y - b) = a, where a and b are constants. Show, by drawing a straight line graph, that this is so. Use your graph to find probable values of a and b. 31 [AEB] An equation can be written in the form x = F(x) and it is known that the equation has only one real root and that this root is near x 1 . Explain, with the aid of a diagram, how, if IF'(x)l < 1, the iterative formula x,+ 1 = F(x,), (r = 1, 2, ... ) will give the root to whatever accuracy is required.