10-Lines and Planes in 3-Space

10-Lines and Planes in 3-Space - Lines and Planes in...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Lines and Planes in 3-space John E. Gilbert, Heather Van Ligten, and Benni Goetz There are many ways of expressing the equations of lines in 2-space (you probably learned the slope-intercept and point-slope formulas among others, for example). What we want now is to do the same for lines and planes in 3-space. Here vectors will be particularly convenient. Lines: Two points determine a line both in 2-space and 3-space . 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 vector-form 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 slope-intercept form for a line in the plane? Sometimes it’s useful to write the equation for a line in coordinate-form : 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 ....
View Full Document

This note was uploaded on 04/12/2011 for the course M 408d taught by Professor Sadler during the Spring '07 term at University of Texas.

Page1 / 6

10-Lines and Planes in 3-Space - Lines and Planes in...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online