Exercises

Question 13.7

What is the domain of ?

Question 13.8

What is the domain of ?

Question 13.9

Evaluate r(2) and r(−1) for .

Question 13.10

Does either of P = (4, 11, 20) or Q = (−1, 6, 16) lie on the path r(t) = 〈1 + t, 2 + t2, t4〉?

Question 13.11

Find a vector parametrization of the line through P = (3, −5, 7) in the direction v = 〈3, 0, 1〉.

Question 13.12

Find a direction vector for the line with parametrization .

Question 13.13

Match the space curves in Figure 8 with their projections onto the xy-plane in Figure 9.

Question 13.14

Match the space curves in Figure 8 with the following vector-valued functions:

  • r1(t) = 〈cos 2t, cos t, sin t〉

  • r2(t) = 〈t, cos 2t, sin 2t〉

  • r3(t) = 〈1, t, t〉

Question 13.15

Match the vector-valued functions (a)–(f) with the space curves (i)–(vi) in Figure 10.

  • r(t) = 〈t + 15, e0.08t cos t, e0.08t sin t〉

  • r(t) = 〈cos t, sin t, sin 12t〉

  • r(t) = 〈cos3 t, sin3 t, sin 2t〉

  • r(t) = 〈t, t2, 2t〉

  • r(t) = 〈cos t, sin t, cos t sin 12t〉

Question 13.16

Which of the following curves have the same projection onto the xy-plane?

  • r1(t) = 〈t, t2, et〉

  • r2(t) = 〈et, t2, t〉

  • r3(t) = 〈t, t2, cos t〉

Question 13.17

Match the space curves (A)–(C) in Figure 11 with their projections (i)–(iii) onto the xy-plane.

Question 13.18

Describe the projections of the circle r(t) = 〈sin t, 0, 4 + cos t〉 onto the coordinate planes.

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In Exercises 13–16, the function r(t) traces a circle. Determine the radius, center, and plane containing the circle.

Question 13.19

r(t) = (9 cos t)i + (9 sin t)j

Question 13.20

r(t) = 7i + (12 cos t)j + (12 sin t)k

Question 13.21

r(t) = 〈sin t, 0, 4 + cos t〉

Question 13.22

r(t) = 〈6 + 3 sin t, 9, 4 + 3 cos t〈

Question 13.23

Let be the curve r(t) = 〈t cos t, t sin t, t〉.

  • Show that lies on the cone x2 + y2 = z2.

  • Sketch the cone and make a rough sketch of on the cone.

Question 13.24

Use a computer algebra system to plot the projections onto the xy- and xz-planes of the curve r(t) = 〈t cos t, t sin t, t〉 in Exercise 17.

In Exercises 19 and 20, let

r(t) = 〈sin t, cos t, sin t cos 2t〉

as shown in Figure 12.

Question 13.25

Find the points where r(t) intersects the xy-plane.

Question 13.26

Show that the projection of r(t) onto the xz-plane is the curve

Question 13.27

Parametrize the intersection of the surfaces

using t = y as the parameter (two vector functions are needed as in Example 3).

Question 13.28

Find a parametrization of the curve in Exercise 21 using trigonometric functions.

Question 13.29

Viviani’s Curve is the intersection of the surfaces (Figure 13)

Viviani’s curve is the intersection of the surfaces x2 + y2 = z2 and y = z2.

  • Parametrize each of the two parts of corresponding to x ≥ 0 and x ≤ 0, taking t = z as parameter.

  • Describe the projection of onto the xy-plane.

  • Show that lies on the sphere of radius 1 with center (0, 1, 0). This curve looks like a figure eight lying on a sphere [Figure 13(B)].

Question 13.30

Show that any point on x2 + y2 = z2 can be written in the form (z cos θ, z sin θ, z) for some θ. Use this to find a parametrization of Viviani’s curve (Exercise 23) with θ as parameter.

Question 13.31

Use sine and cosine to parametrize the intersection of the cylinders x2 + y2 = 1 and x2 + z2 = 1 (use two vector-valued functions). Then describe the projections of this curve onto the three coordinate planes.

Question 13.32

Use hyperbolic functions to parametrize the intersection of the surfaces x2 − y2 = 4 and z = xy.

Question 13.33

Use sine and cosine to parametrize the intersection of the surfaces x2 + y2 = 1 and z = 4x2 (Figure 14).

Intersection of the surfaces x2 + y2 = 1 and z = 4x2.

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In Exercises 28–30, two paths r1(t) and r2(t) intersect if there is a point P lying on both curves. We say that r1(t) and r2(t) collide if r1(t0) = r2(t0) at some time t0.

Question 13.34

Which of the following statements are true?

  • If r1 and r2 intersect, then they collide.

  • If r1 and r2 collide, then they intersect.

  • Intersection depends only on the underlying curves traced by r1 and r2, but collision depends on the actual parametrizations.

Question 13.35

Determine whether r1 and r2 collide or intersect:

Question 13.36

Determine whether r1 and r2 collide or intersect:

In Exercises 31–40, find a parametrization of the curve.

Question 13.37

The vertical line passing through the point (3, 2, 0)

Question 13.38

The line passing through (1, 0, 4) and (4, 1, 2)

Question 13.39

The line through the origin whose projection on the xy-plane is a line of slope 3 and whose projection on the yz-plane is a line of slope 5 (i.e., Δz/Δy = 5)

Question 13.40

The horizontal circle of radius 1 with center (2, −1, 4)

Question 13.41

The circle of radius 2 with center (1, 2, 5) in a plane parallel to the yz-plane

Question 13.42

The ellipse in the xy-plane, translated to have center (9, −4, 0)

Question 13.43

The intersection of the plane with the sphere x2 + y2 + z2 = 1

Question 13.44

The intersection of the surfaces

Question 13.45

The ellipse in the xz-plane, translated to have center (3, 1, 5) [Figure 15(A)]

The ellipses described in Exercises 39 and 40.

Question 13.46

The ellipse , translated to have center (3, 1, 5) [Figure 15(B)]