James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

850 Chapter 13 Vector Functions
The corkscrew shape of the helix in Example 4 is familiar from its occurrence in
coiled springs. It also occurs in the model of DNA (deoxyribonucleic acid, the genetic
material of living cells). In 1953 James Watson and Francis Crick showed that the structure
of the DNA molecule is that of two linked, parallel helixes that are intertwined as in
Figure 3.
In Examples 3 and 4 we were given vector equations of curves and asked for a geometric
description or sketch. In the next two examples we are given a geometric description
of a curve and are asked to find parametric equations for the curve.
FIGURE 3
A double helix
Figure 4 shows the line segment PQ in
Example 5.
Q(2, _1, 3)
z
Example 5 Find a vector equation and parametric equations for the line segment that
joins the point Ps1, 3, 22d to the point Qs2, 21, 3d.
SOLUTION In Section 12.5 we found a vector equation for the line segment that joins
the tip of the vector r 0 to the tip of the vector r 1 :
rstd − s1 2 tdr 0 1 tr 1 0 < t < 1
(See Equation 12.5.4.) Here we take r 0 − k1, 3, 22l and r 1 − k2, 21, 3l to obtain a
vector equation of the line segment from P to Q:
rstd − s1 2 td k1, 3, 22l 1 tk2, 21, 3l 0 < t < 1
x
FIGURE 4
P(1, 3, _2)
y
or rstd − k1 1 t, 3 2 4t, 22 1 5tl 0 < t < 1
The corresponding parametric equations are
x − 1 1 t y − 3 2 4t z − 22 1 5t 0 < t < 1 ■
Example 6 Find a vector function that represents the curve of intersection of the
cylinder x 2 1 y 2 − 1 and the plane y 1 z − 2.
SOLUTION Figure 5 shows how the plane and the cylinder intersect, and Figure 6
shows the curve of intersection C, which is an ellipse.
z
z
y+z=2
(0, _1, 3)
C
(_1, 0, 2)
(1, 0, 2)
≈+¥=1
0
(0, 1, 1)
x
y
x
y
FIGURE 5 FIGURE 6
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