Page 173 - Advanced engineering mathematics

6.1 Vectors in the Plane and 3-Space 153

z

P

y
P 1

P 0
x

FIGURE 6.10 Determining
parametric equations of a line.

Usually we write these equations as
x =−2 + 11t, y =−4 + 5t, z = 7 − 14t.

These are parametric equations of L.As t varies over the real numbers, the point (−2 +
11t,−4 + 5t,7 − 14t) varies over L. We obtain (−2,−4,7) when t = 0 and (9,1,−7) when
t = 1.
The reasoning used in this example can be carried out in general. Suppose we are given
points P 0 :(x 0 , y 0 , z 0 ) and P 1 :(x 1 , y 1 , z 1 ), and we want parametric equations of the line L through
P 0 and P 1 . The vector
(x 1 − x 0 )i + (y 1 − y 0 )j + (z 1 − z 0 )k
is along L, as is the vector
(x − x 0 )i + (y − y 0 )j + (z − z 0 )k

from P 0 to an arbitrary point (x, y, z) on L. These vectors (see Figure 6.10), being both along L,
are parallel, hence for some real t,
(x − x 0 )i + (y − y 0 )j + (z − z 0 )k
= t[(x 1 − x 0 )i + (y 1 − y 0 )j + (z 1 − z 0 )k].

Then
x − x 0 = t(x 1 − x 0 ), y − y 0 = t(y 1 − y 0 ), z − z 0 = t(z 1 − z 0 ).
Parametric equations of the line are
x = x 0 + t(x 1 − x 0 ), y = y 0 + t(y 1 − y 0 ), z = z 0 + t(z 1 − z 0 ),
with t taking on all real values. We get P 0 when t = 0 and P 1 when t = 1.

EXAMPLE 6.1
Find parametric equations of the line through (−1,−1,7) and (7,−1,4).
Let one of these points be P 0 and the other P 1 . To be specific, choose P 0 = (−1,−1,7) =
(x 0 , y 0 , z 0 ) and P 1 = (7,−1,4) = (x 1 , y 1 , z 1 ). The line through these points has parametric
equations

x =−1 + (7 − (−1))t, y =−1 + (−1 − (−1))t, z = 7 + (4 − 7)t

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October 14, 2010 14:21 THM/NEIL Page-153 27410_06_ch06_p145-186