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Lecture 5-2 - Linesand planes:doingit If i s p e rp e n d i...

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Lines and planes: doing it
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Planes described using vectors If is perpendicular to plane and is a point of , we can describe like this: Get a linear equation: Any linear equation gives a plane. u = æ , ! , " ç P u = ( a , b , c ) P P P = {( x , y , z ) : æ x J a , y J b , z J c ç Ğ u = 0} ( x J a ) + ! ( y J b ) + " ( z J c ) = 0
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Planes described using vectors Describe the plane using vectors. Normal vector: . How did I know? Trick: always just use the coefficients of , , and To translate: find one solution by eyeballs. A solution: . So the plane is the set of endpoints of vectors such that Another way to say it: it is what you get when you take the set of vectors perpendicular to and translate them all by (and then just keep the set of endpoints) x J 3 y + 47 z J 28 = 0 æ 1, J 3, 47 ç x y z ( J 16, 1, 1) v ( v J æ J 16, 1, 1 ç ) Ğ æ 1, J 3, 47 ç = 0. æ 1, J 3, 47 ç æ J 16, 1, 1 ç
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Practice Describe the plane using vectors. 3 x J 4 y J 5 z = 6
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Who cares? Using this approach, you can prove that any plane is the set of solutions of a linear equation in (see book!).
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Lecture 5-2 - Linesand planes:doingit If i s p e rp e n d i...

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