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Unformatted text preview: 14. LINES AND PLANES 14.1. Equations of a straight line in space. In our discussion of lines and planes, we will use the notion of a position vector. A position vector is a vector whose initial point is at the origin. Thus, the components of a position vector are the same as the coordinates of its terminal point. To specify a line in space, much like in the case of a line in the xy-plane, we need a point on the line and the direction of the line. If r is the position vector of a point P ( x , y , z ) on the line and if v is a vector colinear with the line, then the position vector r is on the line if and only if r = r + t v for some t ∈ R . (14.1) In other words, as t runs through the real numbers, the position vectors r + t v trace out the line. We call (14.1) a (parametric) vector equation of the line L through P and colinear with v . Writing v in component form, v =〈 a , b , c 〉 and r = (big x , y , z )big , we can express (14.1) in terms of the components of the vectors: (big x , y , z )big = (big x , y , z )big +〈 ta , tb , tc 〉 = (big x + ta , y + tb , z + tc )big , or equivalently: x = x + ta , y = y + tb , z = z + tc ( t ∈ R ). (14.2) These are the parametric (scalar) equations of L . In general, the vector v =〈 a , b , c 〉 is determined up to multiplication by a nonzero scalar, that is, we can use any nonzero scalar multiple of v in place of v . This will change the coefficients in equations (14.1)and (14.2), but not the underlying set of points. We refer to the coordinates a , b , c of any vector v colinear with L as the direction numbers of L . Another common way to specify a line in space are the symmetric equations of L . Given the scalar equations (14.2), with abc negationslash= 0, we can eliminate t to obtain the equations x- x a = y- y b = z- z c . (14.3) These are the symmetric equations of L . If one of a , b , or c is zero—say, b = 0, we obtain the symmetric equations of L by keeping that scalar equation and eliminating t from the other two: y = y , x- x a = z- z c . Finally, if two among a , b , c are equal to zero, the corresponding scalar equations represent the symmetric equations of L after discarding the third parametric equation—say, if a = b = 0, the symmetric equations of L are x = x , y = y . Finally, we can describe a line in space by the coordinates of any two distinct points P 1 ( x 1 , y 1 , z 1 ) and P 2 ( x 2 , y 2 , z 2 ) on the line. Indeed, if P 1 and P 2 are on the line,the vector---→ P 1 P 2 = (big x 2- x 1 , y 2- y 1 , z 2- z 1 )big is colinear with the line, so we can write the symmetric equations (14.3) in the form x- x 1 x 2- x 1 = y- y 1 y 2- y 1 = z- z 1 z 2- z 1 , (14.4) (or something like it, for other types of symmetric equations)....
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This note was uploaded on 06/09/2008 for the course M 408d taught by Professor Sadler during the Summer '07 term at University of Texas at Austin.
- Summer '07