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# 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 Suppose L is a line in space. Let d be a vector parallel to L ; d is called a direction vector. Suppose X 0 and X are any two distinct points on L . Then --→ X 0 X = td , for some parameter t . --→ X 0 X = td is called the vector equation of the line. X 0 L d X 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 = td 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
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 s z = 2 + s do not intersect. Such lines are called skew lines.

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