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= 3i+j-2k and c =i-7j+ 12k are lincarly dependent or linearly independent. Show that the vectors a = 3i-j+4k, b = i- 3j- 2k and c = 4i - 3j +2k are linearly independent. Find numbers a, 3 and such that d = 2i + 3j k can be expressed in the form d = aa +Bb+yc. 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 = d = a +b+ Ac is parallel to the yz-plane. -i+ 2j +4k, find a mumber A such that %3D