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Unformatted text preview: ons of the line that passes through
the points A 2, 4, 3 and B 3, 1, 1 .
(b) At what point does this line intersect the xyplane?
SOLUTION 2 1 P z y0 _1 4
y (a) We are not explicitly given a vector parallel to the line, but observe that the vector v
l
with representation AB is parallel to the line and L v
A 3 2, 1 4, 1 3 1, Thus, direction numbers are a 1, b
5, and c
P0, we see that parametric equations (2) are FIGURE 4 x 2 t y 4 5, 4 4. Taking the point 2, 4, z 5t 3 3 as 4t and symmetric equations (3) are
x 2 y 1 4 z 5 3
4 (b) The line intersects the xyplane when z 0, so we put z
tions and obtain
x2
y4
3
1
5
4
This gives x 11
4 1
4 and y 0 in the symmetric equa , so the line intersects the xyplane at the point ( 11 , 1 , 0).
44 In general, the procedure of Example 2 shows that direction numbers of the line L
through the points P0 x 0 , y0 , z0 and P1 x 1, y1, z1 are x 1 x 0 , y1 y0 , and z1 z0 and so
symmetric equations of L are
x
x1 x0
x0 y
y1 z
z1 y0
y0 z0
z0 Often, we need a description, not of an entire line, but of just a line segment. How, for
instance, could we describe the line segment AB in Example 2? If we put t 0 in the parametric equations in Example 2(a), we get the point 2, 4, 3 and if we put t 1 we get
3, 1, 1 . So the line segment AB is described by the parametric equations
x 2 t y 4 5t z 3 4t 0 t or by the corresponding vector equation
rt 2 t, 4 5 t, 3 4t 0 t 1 1 5E13(pp 858867) 1/18/06 11:28 AM Page 861 SECTION 13.5 EQUATIONS OF LINES AND PLANES ❙❙❙❙ 861 In general, we know from Equation 1 that the vector equation of a line through the (tip
of the) vector r 0 in the direction of a vector v is r r 0 t v. If the line also passes through
(the tip of) r1, then we can take v r1 r 0 and so its vector equation is
r r0 t r1 r0 1 t r0 t r1 The line segment from r 0 to r1 is given by the parameter interval 0
4 x 1 x 5 L™ 1 t r0 0 t r1 t 1 EXAMPLE 3 Show that the lines L 1 and L 2 with parametric equations z L¡ 1. The line segment from r 0 to r1 is given by the vector equation
rt  The lines L 1 and L 2 in Example 3, shown in
Figure 5, are skew lines. t 2s t y
y 2
3 z 3t z s 4 t
3 4s are skew lines; that is, they do not intersect and are not parallel (a...
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This note was uploaded on 02/04/2010 for the course M 56435 taught by Professor Hamrick during the Fall '09 term at University of Texas at Austin.
 Fall '09
 hamrick

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