section33 - Chapter 3 MAT188H1F Lec03 Burbulla Chapter 3...

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Chapter 3 MAT188H1F Lec03 Burbulla Chapter 3 Lecture Notes Fall 2007 Chapter 3 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 3 Chapter 3 3.3: Lines and Planes in 3 Dimensions Chapter 3 Lecture Notes MAT188H1F Lec03 Burbulla
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Chapter 3 3.3: Lines and Planes in 3 Dimensions Vector Equation of a Line I Suppose L is a line in space. I Let ~ d be a vector parallel to L ; ~ d is called a direction vector. I Suppose X 0 and X are any two distinct points on L . I Then --→ X 0 X = t ~ d , for some parameter t . I --→ X 0 X = t ~ d is called the vector equation of the line. ± ± ± ± ± ± ± ± ± ± ± ± X 0 L ± ± ± ±> ~ d X r r Chapter 3 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 3 3.3: Lines and Planes in 3 Dimensions Parametric Equations of a Line Let X 0 have coordinates ( x 0 , y 0 , z 0 ); let X have coordinates ( x , y , z ); let ~ d = a b c . Then --→ X 0 X = t ~ d x - x 0 y - y 0 z - z 0 = t a b c x = x 0 + at y = y 0 + bt z = z 0 + ct These are called parametric equations (or scalar equations) of the line L . Warning: the parametric equations of a line are not unique! Any vector parallel to ~ d could be used, and any other point on L could be used for X 0 . Chapter 3 Lecture Notes MAT188H1F Lec03 Burbulla
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Chapter 3 3.3: Lines and Planes in 3 Dimensions Example 1 Find parametric equations of the line that passes through the two points with coordinates (1 , 3 , - 1) and (2 , 4 , 3) . Solution: We can take X 0 to be either point, say (1 , 3 , - 1) . For the direction vector we can take ~ d = 2 - 1 4 - 3 3 - ( - 1) = 1 1 4 . Thus parametric equations of the line are x = 1 + t y = 3 + t z = - 1 + 4 t , where t is a parameter. Chapter 3 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 3 3.3: Lines and Planes in 3 Dimensions Example 2: Skew Lines Show that the lines L 1 and L 2 with parametric equations L 1 : x = 3 + t y = - 1 + t z = 2 and L 2 : x = s y = 5 - 2
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This note was uploaded on 06/22/2008 for the course ENGINEERIN MAT188 taught by Professor Burbula during the Fall '07 term at University of Toronto- Toronto.

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section33 - Chapter 3 MAT188H1F Lec03 Burbulla Chapter 3...

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