# Algebra Some Current Trends by Avramov L.L. (ed.), Tchakerian K.B. (ed.)

By Avramov L.L. (ed.), Tchakerian K.B. (ed.)

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6. Three non-coplanar vectors u, v, w are given. Referred to these vectors as base vectors, the vectors, e, f, g, h are given by: 1 1 2 —1 e= (0) , f= (1) , g = (3), h = 4 ( 1) . 0 1 3 0 Express e, f, g, h as column vectors taking v +w, w +u, u + v as base vectors. 7. Write down the coordinates of the mid-points of the lines joining the following pairs of points: (i) (2, 4), (4, 6); (ii) (-2, 4), (4, 2); (iii) (— 1, — 3), (2, —4); (iv) (3, 1, —5), (2, —4, 0); (v) (a + b, a, a — b), (a —b, —a, — a — b).

Prove that the position vector of L is c. CM . NB — +1, where the magnitude and sense of each line segment is taken into account. ) 11. If a transversal cuts the sides BC, CA, AB of a triangle ABC at L, M, N respectively, prove that BL. CM. AN _ —1 LC . NB magnitudes and senses of each line segment being taken into account. ) 12. ABCD is a plane quadrilateral. AB and DC meet at P; BC and AD meet at Q. Prove that the mid-points of AC, BD and PQ are collinear. 13. P, Q are variable points on two skew lines.

Why is it not possible to express CD in terms of the two vectors PQ, AR? 13. ABCDEF is a regular hexagon, and AB = a, BC = b. Find CD, DE, EF, FA in terms of a and b. 14. ABCDA'B'C'D' is a cuboid whose base ABCD is a square of side 2 units. , are vertical and of magnitude 4 units. E is the mid-point 31 VECTORS AND VECTOR GEOMETRY [2 of AB, G is the mid-point of B'C' and F is the mid-point of CC'. a, b, c are three vectors, each of magnitude 1 unit in the directions AB, AD, AA' respectively. Find, in terms of a, b, c the vectors DE, AF, EF, GF, GE.