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1. Let a,b and c be three linearly independent (non complanar) vectors. How much should be and to make v and w parallel, if a)v=2a+3b,b)v=a+4b.c)v=a+2ab+cd)v=a+ab+c.w=4a+bw=2a+8b+cw=a+4ab+2cw=2a+2b+c 2. Let a,b and c be three linearly independent (non complanar) vectors. Are vectors r, s,t linearly independent? a) r=3a+2b+c,s=5a3b2c,t=0 b) r=a+b+c,s=2a2b2c,t=2a5b+c c) r=a,s=a+b,t=a+b+c d) r=2a+b+c,s=3a+2b+c,t=a+b 3. Show (with vectors) that the medians of a triangle can form a triangle. 4. Assume, E,F,G denotes the midpoints of the sides of a triangle. Give the position vectors of the vertices with the help of the position vectors of these points. 5. Assume a,b,c and d are the position vectors of points A,B,C,D respectively. Assume a,b,c are non complanar (i.e. they are linearly independent), and d=a+b+c. Show, the points A,B,C,D are in a plane if and only if ++=1. 6. Show (with vectors) that the bisector of a triangle divides the opposite side in the ratio of the neighbouring sides. 7. The coordinates of vectors a,b,c and v with respect to the same basis are given. Express v as a linear combination of a,b and c+if it is possible. a) a=(1,1,0),b=(1,1,0).c=(1,1,1),v=(3,5,7) b) a=(1,0,0),b=(0,1,0).c=(1,1,0),v=(2,1,3) c) a=(1,0,0),b=(1,1,1),c=(0,0,1),v=(3,1,2)

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