If the position vectors of the points A, B, C, D are a = i+j+k, b= 2i + 3j, c = 3i + 5j 2k, d = k -j, show the line segments AB and CD are parallel and find the ratio of their lengths. Given the two vector combinations a+ 4b = 4i + 4j + 6k, a- 2b = i+7j + 3k, find a and b. The position vectors of the points A. B and C are a = 3i -j-k, b= 2i + 2j +7k and c = 5i + 2j- 3k. Find (i) the position vector of the centroid of A, B and C. (ii) the position vectors of the points P and Q which divide AB internally and externally in the ratios AP: PB = 1 : 2 and AQ : QB = -2:1. Determine whet her the vectors a = 2i 3j + 5k, b = 31+j- 2k and c = i- 7j + 12k are linearly dependent or linearly independent. Show that the vectors a = 3i-j+4k, b = i- 3j 2k and e = 4i 3j+2k are linearly independent. Find mumbers a, 3 and 7 such that d 2i + 3j - k can be expressed in the form d= aa +3b+ye. Prove that (i) the three points A(1, 4, 2), B(3, 2, 4) and C(5,0,6) are collinear, (ii) the four points P(2,-1, 1), Q(1,3.-2), R(-2, 1,-3) and S(3,2,0) are coplanar. If a = i+ 3j - k, b 2i + 4j- 2k and c = -i+ 2j + 4k, find a number A such that d = a +b + Ac is parallel to the yz-plane.