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13.5 Flats

# 13.5 Flats - ET 12.5 M 13.5 Lines and Planes Internal...

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ET 12.5; M 13.5. Lines and Planes Internal descriptions. To specify a line, give the position, ~ r 0 say, of some point in the line and give a nonzero vector, ~v say, running along the line. Then the position ~ r of a point in the line satisfies the vector equations ~ r - ~ r 0 = t~v, and ~ r = ~ r 0 + t~v. ( * ) Having two distinct points in the line also leads to such equations. When the parameter t is time, the vector ~v is velocity. In R 3 , the vector equation ( * ) corresponds to three scalar parametric equations. Solve for t in those equations to get symmetric equations . The idea in ( * ) works for a plane in R 3 , but we need two nonparallel vectors lying in the plane. Then ~ r = ~ r 0 + s~u + t~v ( ) with two parameters. Such equations also come from having three nonaligned points in the plane. 1

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ET 12.5; M 13.5. Lines and Planes External descriptions. In equation ( ), the two vectors ~u and ~v implicitly determine how the plane is tilted. In R 3 , it is simpler to specify one nonzero vector, ~n say, perpendicular to the plane. Then ~n ( ~ r - ~ r 0 ) for the position vectors ~ r and ~ r 0
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