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Unformatted text preview: MA1104 Multivariable Calculus Lecture 2 Dr KU Cheng Yeaw Thursday Jan 15, 2008 Recap Last lecture ... • we introduce coordinate systems for threedimensional space. • we introduce vectors and study their geometric and algebraic properties  CauchySchwartz Inequality, Triangle Inequality, dot product, cross product, a · ( b × c ) , a × ( b × c ) ... Overview In this lecture, • using vectors, we will define and study equations for lines and planes. • we will introduce some special type of surfaces (quadric surfaces) other than planes. The notion of traces is useful to sketch these surfaces. Equations of Lines Objective: Define lines in R 3 using vectors. A line L in R 3 is determined when we know two things: • a point P ( x ,y ,z ) on L , and • the direction of L (which can be conveniently represented by a position vector v parallel to L ) Let P ( x,y,z ) denote an arbitrary point on L . Let r and r denote the position vectors of P and P respectively. Let v be a vector parallel to L , so→ P P = t v for some scalar t . Then r = r +→ P P, so that Vector Equation of Line r = r + t v which is a vector equation of L . Each parameter t gives the position vector r of a point on L . As t varies, the line is traced out by the tip of the vector r . We can write the vector equation in the component form: v = h a,b,c i , r = h x ,y ,z i , r = h x,y,z i . Two vectors are equal if and only if the corresponding components are equal. Therefore, we have r = r + t v h x,y,z i = h x ,y ,z i + t h a,b,c i . Parametric Equation of Line x = x + at,y = y + bt,z = z + ct. The numbers a , b and c are called direction numbers of the line L . If none of the direction numbers is , then we can solve each of these equations for t , equate the results, and obtain Symmetric Equation of Line x x a = y y b = z z c If one of the direction numbers is , we still can eliminate t . For example, if a = 0 , b 6 = 0 and c 6 = 0 , then we could write the equations for L as x = x , y y b = z z c . Notice a = 0 implies that v lies on the zyplane and so the xcoordinate of every point on L remains unchanged as t varies. Therefore, L lies on a plane parallel to the zyplane. The vector equation and parametric equations of a line are not unique. If we change the point r or the parameter t or choose a different parallel vector v , then the equations change. Therefore, direction numbers are not unique. Example 1. Find an equation of the line passing through P (1 , 2 , 1) and Q (5 , 3 , 4) . Solution. A vector parallel to the line is→ PQ = h 5 1 , 3 2 , 4 ( 1) i = h 4 , 5 , 5 i . Pick a point on the line, say h 1 , 2 , 1 i . Then the parametric equations for the line are x = 1 + 4 t,y = 2 5 t,z = 1 + 5 t....
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This note was uploaded on 03/19/2012 for the course MATH 1104 taught by Professor Kuchengyeaw during the Spring '08 term at National University of Singapore.
 Spring '08
 Kuchengyeaw
 Algebra, Multivariable Calculus, Vectors

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