This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: The University of Sydney Math3061 Geometry and Topology Web page: www.maths.usyd.edu.au/u/UG/SM/MATH3061/ 2009 Tutorial 1 1. (Revision) ( i ) Given that A = (1 , 1), B = (2 , 3) and C = (4 , 5), find AB , BC and CA . Check your answers by calculating the sum of these three vectors. ( ii ) Find an equation for the line through the point (1 , 1) in the direction given by the vector bracketleftbigg 2 3 bracketrightbigg . Solution. ( i ) AB = bracketleftbigg 2 1 3 1 bracketrightbigg = bracketleftbigg 1 2 bracketrightbigg , BC = bracketleftbigg 4 2 5 3 bracketrightbigg = bracketleftbigg 2 2 bracketrightbigg , CA = bracketleftbigg 1 4 1 5 bracketrightbigg = bracketleftbigg 3 4 bracketrightbigg . This can be checked by finding the sum of AB + BC + CA. We have bracketleftbigg 1 2 bracketrightbigg + bracketleftbigg 2 2 bracketrightbigg + bracketleftbigg 3 4 bracketrightbigg = bracketleftbigg bracketrightbigg . ( ii ) The line ax + by + c = 0 has direction vector bracketleftbigg b a bracketrightbigg . So the line through (1 , 1) in the direction bracketleftbigg 2 3 bracketrightbigg has an equation of the form 3 x 2 y + c = 0 , where 3 2 + c = 0, that is, the equation of the line is 3 x 2 y 1 = 0. 2. Determine (in each case) if the given formula defines as a transformation of the plane, and if it does, find a formula for the inverse transformation 1 . ( i ) ( x, y ) = ( y, x + 2), ( ii ) ( x, y ) = (ln x, y ), ( iii ) ( x, y ) = ( x 2 y, y ), ( iv ) ( x, y ) = ( x, y + x 2 ). Solution. ( i ) Let ( u, v ) be any point in E . Then ( x, y ) = ( u, v ) ( x, y ) = ( v 2 , u ) . That is, given any point ( u, v ) E , there is a unique point ( x, y ), namely ( x, y ) = ( v 2 , u ), such that ( x, y ) = ( u, v ).This shows that is bijective. So is a transformation and 1 ( u, v ) = ( v 2 , u ), or in more usual notation, 1 ( x, y ) = ( y 2 , x ). ( ii ) ( x, y ) is not defined for all ( x, y ) E . Hence is not a transformation of the plane. 2 ( iii ) Let ( u, v ) be any point in E . Then ( x, y ) = ( u, v ) ( x 2 y, y ) = ( u, v ) ( x, y ) = ( u 2 v, v ). This shows that is bijective. So is a transformation and 1 ( x, y ) = ( x 2 y, y ). (In this case, = 1 .) ( iv ) Let ( u, v ) be any point in E . Then ( x, y ) = ( u, v ) ( x, y + x 2 ) = ( u, v ) ( x, y ) = ( u, v u 2 ). This shows that is bijective. So is a transformation and 1 ( x, y ) = ( x, y x 2 )....
View
Full
Document
This note was uploaded on 10/17/2011 for the course MATH 3061 taught by Professor ? during the Three '11 term at University of Sydney.
 Three '11
 ?
 Geometry, Topology

Click to edit the document details