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Chapter 13

Vector Functions

Educators

WZ

Problem 1

Find the domain of the vector function.

$ r(t) = \biggr\langle \ln (t + 1), \frac{t}{\sqrt{9 - t^2}}, 2^t \biggr\rangle $

WZ
Wen Z.
Numerade Educator

Problem 2

Find the domain of the vector function.

$ r(t) = \cos t i + \ln t j + \frac{1}{t - 2}\ k $

WZ
Wen Z.
Numerade Educator

Problem 3

Find the limit.

$ \lim_{t\to 0} \left(e^{-3t} i + \frac{t^2}{\sin^2 t}\ j + \cos 2t\ k \right) $

Bobby B.
University of North Texas

Problem 4

Find the limit.

$ \lim_{t\to 1} \left(\frac{t^2 - t}{t - 1}\ i + \sqrt{t + 8}\ j + \frac{\sin \pi t}{\ln t}\ k \right) $

WZ
Wen Z.
Numerade Educator

Problem 5

Find the limit.

$ \lim_{t\to\infty} \biggr\langle\frac{1 + t^2}{1 - t^2}, \tan^{-1} t , \frac{1 - e^{-2t}}{t} \biggr\rangle $

WZ
Wen Z.
Numerade Educator

Problem 6

Find the limit.

$ \lim_{t\to\infty} \biggr\langle te^{-t} , \frac{t^3 + t}{2t^3 - 1}, t \sin \frac{1}{t} \biggr\rangle $

WZ
Wen Z.
Numerade Educator

Problem 7

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $ t $ increases.

$ r(t) = \langle \sin t , t \rangle $

WZ
Wen Z.
Numerade Educator

Problem 8

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $ t $ increases.

$ r(t) = \langle t^2 - 1 , t \rangle $

WZ
Wen Z.
Numerade Educator

Problem 9

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $ t $ increases.

$ r(t) = \langle t , 2 - t , 2t \rangle $

WZ
Wen Z.
Numerade Educator

Problem 10

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $ t $ increases.

$ r(t) = \langle \sin \pi t , t , \cos \pi t \rangle $

Carson M.
Numerade Educator

Problem 11

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $ t $ increases.

$ r(t) = \langle 3 , t , 2 - t^2 \rangle $

WZ
Wen Z.
Numerade Educator

Problem 12

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $ t $ increases.

$ r(t) = 2 \cos t i + 2 \sin t j + k $

Carson M.
Numerade Educator

Problem 13

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $ t $ increases.

$ r(t) = t^2 i + t^4 j + t^6 k $

WZ
Wen Z.
Numerade Educator

Problem 14

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $ t $ increases.

$ r(t) = \cos t i - \cos t j + \sin t k $

WZ
Wen Z.
Numerade Educator

Problem 15

Draw the projections of the curve on the three coordinate planes. Use these projections to help sketch the curve.

$ r(t) = \langle t , \sin t , 2 \cos t \rangle $

Carson M.
Numerade Educator

Problem 16

Draw the projections of the curve on the three coordinate planes. Use these projections to help sketch the curve.

$ r(t) = \langle t , t , t^2 \rangle $

WZ
Wen Z.
Numerade Educator

Problem 17

Find a vector equation and parametric equations for the line segment that joins $ P $ to $ Q $.

$ P (2, 0, 0), Q (6, 2, -2) $

Carson M.
Numerade Educator

Problem 18

Find a vector equation and parametric equations for the line segment that joins $ P $ to $ Q $.

$ P (-1, 2, -2), Q (-3, 5, 1) $

Carson M.
Numerade Educator

Problem 19

Find a vector equation and parametric equations for the line segment that joins $ P $ to $ Q $.

$ P (0, -1, 1), Q (\frac{1}{2}, \frac{1}{3}, \frac{1}{4}) $

WZ
Wen Z.
Numerade Educator

Problem 20

Find a vector equation and parametric equations for the line segment that joins $ P $ to $ Q $.

$ P (a, b, c), Q (u, v, w) $

WZ
Wen Z.
Numerade Educator

Problem 21

Match the parametric equations with the graphs (labeled I-VI). Give reasons for your choices.

$ x = t \cos t $ , $ y = t $ , $ z = t \sin t $ , $ t \ge 0 $

WZ
Wen Z.
Numerade Educator

Problem 22

Match the parametric equations with the graphs (labeled I-VI). Give reasons for your choices.

$ x = \cos t $ , $ y = \sin t $ , $ z = \frac{1}{(1 + t^2)} $

WZ
Wen Z.
Numerade Educator

Problem 23

Match the parametric equations with the graphs (labeled I-VI). Give reasons for your choices.

$ x = t $ , $ y = \frac{1}{(1 + t^2)} $ , $ z = t^2 $

Carson M.
Numerade Educator

Problem 24

Match the parametric equations with the graphs (labeled I-VI). Give reasons for your choices.

$ x = \cos t $ , $ y = \sin t $ , $ z = \cos 2t $

WZ
Wen Z.
Numerade Educator

Problem 25

Match the parametric equations with the graphs (labeled I-VI). Give reasons for your choices.

$ x = \cos 8t $ , $ y = \sin 8t $ , $ z = e^{0.8t} $ , $ t \ge 0 $

Carson M.
Numerade Educator

Problem 26

Match the parametric equations with the graphs (labeled I-VI). Give reasons for your choices.

$ x = \cos^2 t $ , $ y = \sin^2 t $ , $ z = t $

WZ
Wen Z.
Numerade Educator

Problem 27

Show that the curve with parametric equations $ x = t \cos t $, $ y = t \sin t $, $ z = t $ lies on the cone $ z^2 = x^2 + y^2 $, and use this fact to help sketch the curve.

Carson M.
Numerade Educator

Problem 28

Show that the curve with parametric equations $ x = \sin t $, $ y = \cos t $, $ z = \sin^2 t $ is the curve of intersection of the surfaces $ z = x^2 $ and $ x^2 + y^2 = 1 $. Use this fact to help sketch the curve.

WZ
Wen Z.
Numerade Educator

Problem 29

Find three different surfaces that contain the curve
$ r(t) = 2t i + e^t j + e^{2t} k $

Carson M.
Numerade Educator

Problem 30

Find three different surfaces that contain the curve
$ r(t) = t^2 i + \ln tj + (\frac{1}{t}) k $.

Carson M.
Numerade Educator

Problem 31

At what points does the curve $ r(t) = ti + (2t - t^2) k $ intersect the paraboloid $ z = x^2 + y^2 $?

Carson M.
Numerade Educator

Problem 32

At what points does the helix $ r(t) = \langle \sin t, \cos t, t \rangle $ intersect the sphere $ x^2 + y^2 + z^2 = 5 $?

WZ
Wen Z.
Numerade Educator

Problem 33

Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve.

$ r(t) = \langle \cos t \sin 2t, \sin t \sin 2t, \cos 2t \rangle $

WZ
Wen Z.
Numerade Educator

Problem 34

Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve.

$ r(t) = \langle te^t, e^{-t}, t \rangle $

WZ
Wen Z.
Numerade Educator

Problem 35

Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve.

$ r(t) = \langle \sin 3t \cos t, \frac{1}{4} t, \sin 3t \sin t \rangle $

WZ
Wen Z.
Numerade Educator

Problem 36

Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve.

$ r(t) = \langle \cos (8 \cos t) \sin t, \sin (8 \cos t) \sin t, \cos t \rangle $

WZ
Wen Z.
Numerade Educator

Problem 37

Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve.

$ r(t) = \langle \cos 2t, \cos 3t, \cos 4t \rangle $

WZ
Wen Z.
Numerade Educator

Problem 38

Graph the curve with parametric equations $ x = \sin t $, $ y = \sin 2t $, $ z = \cos 4t $. Explain its shape by graphing its projections onto the three coordinate planes.

WZ
Wen Z.
Numerade Educator

Problem 39

Graph the curve with parametric equations
$$ x = (1 + \cos 16t) \cos t $$
$$ y = (1 + \cos 16t) \sin t $$
$$ z = 1 + \cos 16t $$
Explain the appearance of the graph by showing that it lies on a cone.

WZ
Wen Z.
Numerade Educator

Problem 40

Graph the curve with parametric equations
$$ x = \sqrt{1 - 0.25 \cos^2 10t} \cos t $$
$$ y = \sqrt{1 - 0.25 \cos^2 10t} \sin t $$
$$ z = 0.5 \cos 10t $$
Explain the appearance of the graph by showing that it lies on a sphere.

WZ
Wen Z.
Numerade Educator

Problem 41

Show that the curve with parametric equations $ x = t^2 $, $ y = 1 - 3t $, $ z = 1 + t^3 $ passes through the points $ (1, 4, 0) $ and $ (9, -8, 28) $ but not through the point $ (4, 7, -6) $.

WZ
Wen Z.
Numerade Educator

Problem 42

Find a vector function that represents the curve of intersection of the two surfaces.

The cylinder $ x^2 + y^2 = 4 $ and the surface $ z = xy $

WZ
Wen Z.
Numerade Educator

Problem 43

Find a vector function that represents the curve of intersection of the two surfaces.

The cone $ z = \sqrt{x^2 + y^2} $ and the plane $ z = 1 + y $

WZ
Wen Z.
Numerade Educator

Problem 44

Find a vector function that represents the curve of intersection of the two surfaces.

The paraboloid $ z = 4x^2 + y^2 $ and the parabolic cylinder $ y = x^2 $

Madi S.
Numerade Educator

Problem 45

Find a vector function that represents the curve of intersection of the two surfaces.

The hyperboloid $ z = x^2 - y^2 $ and the cylinder $ x^2 + y^2 = 1 $

Carson M.
Numerade Educator

Problem 46

Find a vector function that represents the curve of intersection of the two surfaces.

The semiellipsoid $ x^2 + y^2 + 4z^2 = 4, y \ge 0 $, and the cylinder $ x^2 + z^2 = 1 $

Carson M.
Numerade Educator

Problem 47

Try to sketch by hand the curve of intersection of the circular cylinder $ x^2 + y^2 = 4 $ and the parabolic cylinder $ z = x^2 $. Then find parametric equations for this curve and use these equations and a computer to graph the curve.

WZ
Wen Z.
Numerade Educator

Problem 48

Try to sketch by hand the curve of intersection of the parabolic cylinder $ y = x^2 $ and the top half of the ellipsoid $ x^2 + 4y^2 + 4z^2 = 16 $. Then find parametric equations for this curve and use these equations and a computer to graph the curve.

Carson M.
Numerade Educator

Problem 49

If two objects travel through space along two different curves, it's often important to know whether they will collide. (Will a missile hit its moving target? Will two aircraft collide?) The curves might intersect, but we need to know whether the objects are in the same position at the same time. Suppose the trajectories of two particles are given by the vector functions
$ r_1 (t) = \langle t^2, 7t - 12, t^2 \rangle $ $ r_2 (t) = \langle 4t - 3, t^2, 5t - 6 \rangle $
for $ t \ge 0 $. Do the particles collide?

Carson M.
Numerade Educator

Problem 50

Two particles travel along the space curves
$ r_1 (t) = \langle t, t^2, t^3 \rangle $ $ r_2 (t) = \langle 1 + 2t, 1 + 6t, 1 + 14t \rangle $
Do the particles collide? Do their paths intersect?

WZ
Wen Z.
Numerade Educator

Problem 51

(a) Graph the curve with parametric equations
$$ x = \frac{27}{26} \sin 8t - \frac{8}{39} \sin 18t $$
$$ y = -\frac{27}{26} \cos 8t + \frac{8}{39} \cos 18t $$
$$ z = \frac{144}{65} \sin 5t $$
(b) Show that the curve lies on the hyperboloid of one sheet
$ 144x^2 + 144y^2 - 25z^2 = 100 $.

Carson M.
Numerade Educator

Problem 52

The view of the trefoil knot shown in Figure 8 is accurate, but it doesn't reveal the whole story. Use the parametric equations
$$ x = (2 + \cos 1.5t) \cos t $$
$$ y = (2 + \cos 1.5t) \sin t $$
$$ z = \sin 1.5t $$
to sketch the curve by hand as viewed from above, with gaps indicating where the curve passes over itself. Start by showing that the projection of the curve onto the $ xy $-plane has polar coordinates $ r = 2 + \cos 1.5t $ and $ \theta = t $, so $ r $ varies between 1 and 3. Then show that $ z $ has maximum and minimum values when the projection is halfway between $ r = 1 $ and $ r = 3 $.
When you have finished your sketch, use a computer to draw the curve with viewpoint directly above and compare with your sketch. Then use the computer to draw the curve from several other viewpoints. You can get a better impression of the curve if you plot a tube with radius 0.2 around the curve. (Use the tubeplot command in Maple or the tubecurve or Tube command in Mathematica.)

WZ
Wen Z.
Numerade Educator

Problem 53

Suppose $ u $ and $ v $ are vector functions that possess limits as $ t \to a $ and let $ c $ be a constant. Prove the following properties of limits.
(a) $ \displaystyle \lim_{t \to a} [u(t) + v(t)] = \displaystyle \lim_{t \to a} u(t) + \displaystyle \lim_{t \to a} v(t) $
(b) $ \displaystyle \lim_{t \to a} cu(t) = c \displaystyle \lim_{t \to a} u(t) $
(c) $ \displaystyle \lim_{t \to a} [u(t) \cdot v(t)] = \displaystyle \lim_{t \to a} u(t) \cdot \displaystyle \lim_{t \to a} v(t) $
(d) $ \displaystyle \lim_{t \to a} [u(t) \times v(t)] = \displaystyle \lim_{t \to a} u(t) \times \displaystyle \lim_{t \to a} v(t) $

WZ
Wen Z.
Numerade Educator

Problem 54

Show that $ \displaystyle \lim_{t \to a} r(t) = b $ if and only if for every $ \varepsilon > 0 $ there is a number $ \delta > 0 $ such that
if $ 0 < \mid t - a \mid < \delta $ then $ \mid r(t) - b \mid < \varepsilon $

WZ
Wen Z.
Numerade Educator