# Ex 12.2,2 - Chapter 12 Class 11 Introduction to Three Dimensional Geometry

Last updated at May 29, 2018 by Teachoo

Last updated at May 29, 2018 by Teachoo

Transcript

Ex 12.2,2 Show that the points P ( 2, 3, 5), Q (1, 2, 3) and R (7, 0, 1) are collinear. If three points are collinear, then they lie on a line. Let first calculate distance between the 3 points i.e. PQ. QR and PR Calculating PQ P ( 2, 3, 5) Q (1, 2, 3) Hence , PQ = x2 x1 2+ y2 y1 2+ z2 z1 2 Here , x1 = 2, y1 = 3, z1 = 5 x2 = 1, y2 = 2, z2 = 3 PQ = 1 2 2+ 2 3 2+ 3 5 2 = 1+2 2+ 2 3 2+ 3 5 2 = 32+ 1 2+( 2)2 = 9+ 1 2+ 2 2 = 9+1+4 = 14 Calculating QR Q ( 1, 2, 3) R (7, 0, 1) QR = x2 x1 2+ y2 y1 2+ z2 z1 2 Here x1 = 1, y1 = 2, z1 = 2 x2 = 7, y2 = 0, z2 = 1 QR = 7 1 2+ 0 2 2+ 1 3 2 = 6 2+ 2 2+ 4 2 = 36+4+16 = 56 = 14 2 2 = 2 14 Calculating PR P ( 2, 3, 5) R (7, 0, 1) PR = x2 x1 2+ y2 y1 2+ z2 z1 2 Here, x1 = 2, y1 = 3, z1 = 5 x2 = 7, y2 = 0, z2 = 1 PR = 7 ( 2) 2+ 0 3 2+ 1 5 2 = 7+2 2+ 3 2+ 6 2 = 9 2+9+36 = 81+9+36 = 126 = 14 3 3 = 3 14 Thus, PQ = 14 , QR = 2 14 & PR = 3 14 So, PQ + QR = 14 + 2 14 = 3 14 = PR Thus, PQ + QR = PR So, if we draw the points on a graph, with PQ + QR = PR We see that points P,Q,R lie on the same line. Thus, P, Q and R all collinear

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