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Unformatted text preview: Lines and Planes in 3space John E. Gilbert, Heather Van Ligten, and Benni Goetz There are many ways of expressing the equations of lines in 2space (you probably learned the slopeintercept and pointslope formulas among others, for example). What we want now is to do the same for lines and planes in 3space. Here vectors will be particularly convenient. Lines: Two points determine a line both in 2space and 3space . So imagine a laser pointer (or a light saber for Star Wars fans) at one of the two points, say b , and shine it towards the other point, say a . If we extend the laser pointer or light saber in both directions, we get a line. To write this as an equation, we just need to write the light saber and the idea of “extending” it mathematically. Represent the light saber as a displacement vector v shown in dark blue, with tail at b and head at a ; so v = a − b . To extend v in both directions we scale the vector by writing t v , shown in lighter blue, where t is a real number. Doing this for all such t gives us the complete line. So in vectorform each point on the line is given by r ( t ) = t v + b . x y z v b a t v + b Isn’t this like the slopeintercept form for a line in the plane? Sometimes it’s useful to write the equation for a line in coordinateform : if we write r ( t ) = ( x ( t ) , y ( t ) , z ( t ) ) , v = ( k, m, n ) , b = ( x 1 , y 1 , z 1 ) , then the vector equation becomes r ( t ) = ( tk + x 1 , tm + x 2 , tn + b 3 ) , giving a second equation for a line: x ( t ) = tk + x 1 , y ( t ) = tm + y 1 , z ( t ) = tn + z 1 . Solving for t in these equations (and writing x instead of x ( t ) and so on), we then get t = x − x 1 k , t = y − y 1 m , t = z − z 1 n , giving a third equation x − x 1 k = y − y 1 m = z − z 1 n . Which of these three equation forms for a line in 3space we use depends on how things are set up, but it’s often simpler to start with the vector form....
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This note was uploaded on 06/05/2011 for the course MATH 408 D taught by Professor Gilbert during the Spring '11 term at University of Texas at Austin.
 Spring '11
 Gilbert
 Equations, Formulas, Slope

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