lecture5-09[1] - Math 18.02(Spring 2009 Lecture 5 Planes...

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Math 18.02 (Spring 2009): Lecture 5 Planes. Parametric equations of curves and lines February 12 Reading Material: From Simmons : 17.1 and 17.2. Last time: Square Systems. Word problem. How many solutions? Equations of planes Today: Planes. Parametric equations of curves and lines 2 Equations of Planes In this section we recall the analytic description of a plane through a point P 0 = ( x 0 , y 0 , z 0 ) and perpendicular to a vector N = ( a, b, c ). This plane is described by all points P = ( x, y, z ) such that the vector --→ P 0 P = ( x - x 0 , y - y 0 , z - z 0 ) is perpendicular to the given vector N : N --→ P 0 P ⇐⇒ N · --→ P 0 P = 0 . Since N · --→ P 0 P = a ( x - x 0 ) + b ( y - y 0 ) + c ( z - z 0 ) = 0 , it follows that the ( analytic ) equation for our plane is a ( x - x 0 ) + b ( y - y 0 ) + c ( z - z 0 ) = 0 . (2.1) 1
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Exercise 1. Find a plane containing P 1 = (1 , 2 , 3) , P 2 = (1 , 4 , 4) and P 3 = (0 , 2 , 6) . Method: Find a vector N normal to two vectors ---→ P 1 P 2 and ---→ P 1 P 3 by computing N = ---→ P 1 P 2 × ---→ P 1 P 3 . Use N and any of P 1 , P 2 , P 3 to write the appropriate equation like in (2.1). Now the details: To write equation (2.1) we take the vector N we just computed and the point P 2 = (1 , 4 , 4) for example 1 and we get that can also be written as 6 x - y + 2 z = 10. Let’s now check if
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