Slide_02 - MA1104 Multivariable Calculus Lecture 2 Dr KU...

Info iconThis preview shows pages 1–14. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MA1104 Multivariable Calculus Lecture 2 Dr KU Cheng Yeaw Thursday Jan 15, 2008 Recap Last lecture ... • we introduce coordinate systems for three-dimensional space. • we introduce vectors and study their geometric and algebraic properties - Cauchy-Schwartz 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 zy-plane and so the x-coordinate of every point on L remains unchanged as t varies. Therefore, L lies on a plane parallel to the zy-plane. 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....
View Full Document

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.

Page1 / 96

Slide_02 - MA1104 Multivariable Calculus Lecture 2 Dr KU...

This preview shows document pages 1 - 14. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online