**814** **CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE**

**1.** Homework Hints available at stewartcalculus.com

**1.** Suppose you start at the origin, move along the -axis a
distance of 4 units in the positive direction, and then move
downward a distance of 3 units. What are the coordinates
of your position?

**2.** Sketch the points , , , and

on a single set of coordinate axes.

**3.** Which of the points , , and

is closest to the -plane? Which point lies in the -plane?

**4.** What are the projections of the point (2, 3, 5) on the -, -,
and -planes? Draw a rectangular box with the origin and

as opposite vertices and with its faces parallel to the coordinate planes. Label all vertices of the box. Find the length of the diagonal of the box.

**5.** Describe and sketch the surface in represented by the equa-
tion .

**6.** (a) What does the equation represent in ? What does
it represent in ? Illustrate with sketches.

(b) What does the equation represent in ? What does represent? What does the pair of equations , represent? In other words, describe the set of points

such that and . Illustrate with a sketch.

**7–8** Find the lengths of the sides of the triangle . Is it a right
triangle? Is it an isosceles triangle?

**7.** , ,

**8.** , ,

*x*

共0, 5, 2兲 共4, 0, ⫺1兲 共2, 4, 6兲 共1, ⫺1, 2兲

*A*共⫺4, 0, ⫺1兲

*yz* *xz*

*B*共3, 1, ⫺5兲 *C*共2, 4, 6兲

*xy yz*
*xz*

共2, 3, 5兲

⺢^{3}
*x⫹ y 苷 2*

*x*苷 4 ⺢^{2}

⺢^{3}

*y*苷 3 ⺢^{3}

*z*苷 5 *y*苷 3

*z*苷 5

*共x, y, z兲* *y*苷 3 *z*苷 5

*PQR*

*P共3, ⫺2, ⫺3兲 Q共7, 0, 1兲 R共1, 2, 1兲*
*P共2, ⫺1, 0兲 Q共4, 1, 1兲 R共4, ⫺5, 4兲*

**9.** Determine whether the points lie on straight line.

(a) , ,

(b) , ,

**10.** Find the distance from to each of the following.

(a) The -plane (b) The -plane (c) The -plane (d) The -axis (e) The -axis (f ) The -axis

**11.** Find an equation of the sphere with center and
radius 5. What is the intersection of this sphere with the

-plane?

**12.** Find an equation of the sphere with center and
radius 5. Describe its intersection with each of the coordinate
planes.

**13.** Find an equation of the sphere that passes through the point
and has center .

**14.** Find an equation of the sphere that passes through the origin
and whose center is .

**15–18** Show that the equation represents a sphere, and find its
center and radius.

**15.**

**16.**

**17.**

**18.**

*D共0, ⫺5, 5兲 E共1, ⫺2, 4兲 F共3, 4, 2兲*
*C*共1, 3, 3兲
*B*共3, 7, ⫺2兲

*A*共2, 4, 2兲

共2, ⫺6, 4兲

共4, 3, ⫺1兲 共3, 8, 1兲

共1, 2, 3兲

*x*^{2}*⫹ y*^{2}*⫹ z*^{2}*⫺ 2x ⫺ 4y ⫹ 8z 苷 15*
*x*^{2}*⫹ y*^{2}*⫹ z*^{2}*⫹ 8x ⫺ 6y ⫹ 2z ⫹ 17 苷 0*
*2x*^{2}*⫹ 2y*^{2}*⫹ 2z*^{2}*苷 8x ⫺ 24z ⫹ 1*
*3x*^{2}*⫹ 3y*^{2}*⫹ 3z*^{2}*苷 10 ⫹ 6y ⫹ 12z*

共3, 7, ⫺5兲

*xy* *yz*

*xz* *x*

*y* *z*

共1, ⫺4, 3兲
*xz*

**12.1** **Exercises**

### Comparing this equation with the standard form, we see that it is the equation of a

### sphere with center and radius .

### What region in is represented by the following inequalities?

SOLUTION

### The inequalities

### can be rewritten as

### so they represent the points whose distance from the origin is at least 1 and at most 2. But we are also given that *, so the points lie on or below the xy-plane. *

### Thus the given inequalities represent the region that lies between (or on) the spheres and *and beneath (or on) the xy-plane. It is sketched* in Figure 13.

### 共⫺2, 3, ⫺1兲 s 8 苷 2s2

### ⺢

^{3}

### 1 *艋 x*

^{2}

*⫹ y*

^{2}

*⫹ z*

^{2}

### 艋 4

*z*

### 艋 0

### 1 *艋 x*

^{2}

*⫹ y*

^{2}

*⫹ z*

^{2}

### 艋 4

### 1 *艋 sx*

^{2}

*⫹ y*

^{2}

*⫹ z*

^{2 }

### 艋 2 *共x, y, z兲*

*z*

### 艋 0

*x*

^{2}

*⫹ y*

^{2}

*⫹ z*

^{2}

### 苷 1

*x*

^{2}

*⫹ y*

^{2}

*⫹ z*

^{2}

### 苷 4

**EXAMPLE 7**

**FIGURE 13**
0
1
2

z

x y

98845_ch12_ptg01_hr_809-817.qk_98845_ch12_ptg01_hr_809-817 8/18/11 3:20 PM Page 814

**SECTION 12.2 VECTORS** **815**
**19.** (a) Prove that the midpoint of the line segment from

to is

(b) Find the lengths of the medians of the triangle with ver tices

, , and .

**20.** Find an equation of a sphere if one of its diameters has end -
points and .

**21.** Find equations of the spheres with center that touch
(a) the -plane, (b) the -plane, (c) the -plane.

**22.** Find an equation of the largest sphere with center (5, 4, 9) that
is contained in the first octant.

**23–34** Describe in words the region of represented by the equa-
tions or inequalities.

**23.** **24.**

**25.** **26.**

**27.** **28.**

**29.** , _{30.}

**31.** **32.**

**33.** **34.**

**35–38** Write inequalities to describe the region.

**35.** The region between the -plane and the vertical plane
**36.** The solid cylinder that lies on or below the plane and on

or above the disk in the -plane with center the origin and radius 2

**37.** The region consisting of all points between (but not on) the
spheres of radius and centered at the origin, where
**38.** The solid upper hemisphere of the sphere of radius 2 centered

at the origin

**39.** The figure shows a line in space and a second line

which is the projection of on the -plane. (In other words,

## 冉

^{x}^{1}

^{⫹ x}^{2}

^{2}

^{, }

^{ y}^{1}

^{⫹ y}^{2}

^{2}

^{, }

^{z}^{1}

^{⫹ z}^{2}

^{2}

## 冊

*A共1, 2, 3兲 B共⫺2, 0, 5兲* *C*共4, 1, 5兲
共2, 1, 4兲 共4, 3, 10兲

共2, ⫺3, 6兲

*xy* *yz* *xz*

⺢^{3}

*x*苷 5 *y*苷 ⫺2

*y*⬍ 8 *x*艌 ⫺3

0*艋 z 艋 6* *z*^{2}苷 1

*x*^{2}*⫹ y*^{2}*苷 4 z 苷 ⫺1* *y*^{2}*⫹ z*^{2}苷 16
*x*^{2}*⫹ y*^{2}*⫹ z*^{2}艋 3 *x苷 z*

*x*^{2}*⫹ z*^{2}艋 9 *x*^{2}*⫹ y*^{2}*⫹ z*^{2}*⬎ 2z*

*yz* *x*苷 5

*z*苷 8
*xy*

*r* *R* *r⬍ R*

*L*1 *L*2,

*L*1 *xy*
*P*2*共x*^{2}*, y*2*, z*2兲
*P*1*共x*^{1}*, y*1*, z*1兲

the points on are directly beneath, or above, the points on .)

(a) Find the coordinates of the point on the line . (b) Locate on the diagram the points , , and , where

the line intersects the -plane, the -plane, and the -plane, respectively.

**40.** Consider the points such that the distance from to
is twice the distance from to . Show
that the set of all such points is a sphere, and find its center and
radius.

**41.** Find an equation of the set of all points equidistant from the

points and . Describe the set.

**42.** Find the volume of the solid that lies inside both of the spheres

and

**43.** Find the distance between the spheres and
.

**44.** Describe and sketch a solid with the following properties.

When illuminated by rays parallel to the -axis, its shadow is a circular disk. If the rays are parallel to the -axis, its shadow is a square. If the rays are parallel to the -axis, its shadow is an isosceles triangle.

*L*2

*L*1

*P* *L*1

*A B* *C*

*L*1 *xy* *yz*

*xz*

*P* *P*

*A*共⫺1, 5, 3兲 *P* *B*共6, 2, ⫺2兲

*A*共⫺1, 5, 3兲 *B*共6, 2, ⫺2兲

*x*^{2}*⫹ y*^{2}*⫹ z*^{2}*⫹ 4x ⫺ 2y ⫹ 4z ⫹ 5 苷 0*
*x*^{2}*⫹ y*^{2}*⫹ z*^{2}苷 4

*x*^{2}*⫹ y*^{2}*⫹ z*^{2}苷 4
*x*^{2}*⫹ y*^{2}*⫹ z*^{2}*苷 4x ⫹ 4y ⫹ 4z ⫺ 11*

*z*
*y*
*x*
x

0 z

y 1

1 1

L¡

L™

P

**The term vector is used by scientists to indicate a quantity (such as displacement or veloc-** ity or force) that has both magnitude and direction. A vector is often represented by an arrow or a directed line segment. The length of the arrow represents the magnitude of the vector and the arrow points in the direction of the vector. We denote a vector by printing a letter in boldface or by putting an arrow above the letter

### For instance, suppose a particle moves along a line segment from point to point . **The corresponding displacement vector , shown in Figure 1, has initial point** (the tail) **and terminal point** (the tip) and we indicate this by writing

*AB*

### l

### . Notice that the vec-

**共v兲** 共

^{v}^{l}

### 兲.

*A* *B*

**v** *A*

**v**

### 苷

*B*

**12.2** **Vectors**

**FIGURE 1**
Equivalent vectors
A

B
**v**

C

D
**u**

98845_ch12_ptg01_hr_809-817.qk_98845_ch12_ptg01_hr_809-817 8/18/11 3:20 PM Page 815

**822** **CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE**

**1.** Homework Hints available at stewartcalculus.com

**1.** Are the following quantities vectors or scalars? Explain.

(a) The cost of a theater ticket (b) The current in a river

(c) The initial flight path from Houston to Dallas (d) The population of the world

**2.** What is the relationship between the point (4, 7) and the
vector ? Illustrate with a sketch.

**3.** Name all the equal vectors in the parallelogram shown.

**4.** Write each combination of vectors as a single vector.

*(a) PQ*l
*QR*l

*(b) RP*l
*PS*l

### (c)

*QS*l

*PS*l

*(d) RS*l
*SP*l

*PQ*l
具4, 7典

*B*
*E*

*A*

*D* *C*

*Q*

*R* *S*

*P*

**5.** Copy the vectors in the figure and use them to draw the
following vectors.

(a) (b)

(c) (d)

(e) (f )

**6.** Copy the vectors in the figure and use them to draw the
following vectors.

(a) (b)

(c) (d)

(e) (f )

**7.** In the figure, the tip of and the tail of are both the midpoint
of . Express and in terms of and .

**u v** **u w**

**v w** **u v**

**v** **w**
**u**

**v u w** **u w v**

**a b** **a b**

1

2**a** **3b**

**a 2b** **2b a**

**b** **a**

**c** **d**

*QR* **c** **d** **a** **b**

**b**

**a** **c**

**d**
P

Q

R

**12.2** **Exercises**

### Equating components, we get

### Solving the first of these equations for and substituting into the second, we get

### So the magnitudes of the tensions are

### and

### Substituting these values in and , we obtain the tension vectors

### ⱍ

^{T}^{1}

### ⱍ ^{cos 50} ^{ } ⱍ

^{T}^{2}

### ⱍ ^{cos 32} ^{ 苷 0} ⱍ

^{T}^{1}

### ⱍ ^{sin 50} ^{ } ⱍ

^{T}^{2}

### ⱍ ^{sin 32} ^{ 苷 980}

### ⱍ

^{T}^{2}

### ⱍ ⱍ

^{T}^{1}

### ⱍ ^{sin 50} ^{ } ⱍ

^{T}^{1}

### ⱍ ^{cos 50} ^{}

### cos 32 sin 32 苷 980

### ⱍ

^{T}^{1}

### ⱍ ^{苷} _{sin 50} tan 32 cos 50 ^{980} ⬇ 839 N

### ⱍ

^{T}^{2}

### ⱍ ^{苷} ⱍ

^{T}^{1}

### ⱍ ^{cos 50} ^{}

### cos 32 ⬇ 636 N

### 5 6

**T**1

**⬇ 539 i 643 j**

**T**2

**⬇ 539 i 337 j**

98845_ch12_ptg01_hr_818-827.qk_98845_ch12_ptg01_hr_818-827 8/18/11 3:26 PM Page 822

**SECTION 12.2 VECTORS** **823**
**8.** If the vectors in the figure satisfy and

, what is ?

**9–14** Find a vector with representation given by the directed line
*segment AB*l

*. Draw AB*l

and the equivalent representation starting at the origin.

**9.** , ** _{10.}** ,

**11.** , ** _{12.}** ,

**13.** , ** _{14.}** ,

**15–18** Find the sum of the given vectors and illustrate
geometrically.

**15.** , ** _{16.}** ,

**17.** , ** _{18.}** ,

**19–22** **Find a b, 2a 3b, , **and .

**19.** ,

**20.** ,

**21.** ,

**22.** ,

**23–25** Find a unit vector that has the same direction as the given
vector.

**23.** **24.**

**25.**

**26.** Find a vector that has the same direction as but has
length 6.

**27–28** What is the angle between the given vector and the positive
direction of the -axis?

**27.** **28.**

**29.** If lies in the first quadrant and makes an angle with the
positive -axis and , find in component form.

**30.** If a child pulls a sled through the snow on a level path with a
force of 50 N exerted at an angle of above the horizontal,
find the horizontal and vertical components of the force.

**31.** A quarterback throws a football with angle of elevation and
speed . Find the horizontal and vertical components of
the velocity vector.

### ⱍ

^{u}### ⱍ

^{苷}

### ⱍ

^{v}### ⱍ

^{苷 1}

**u v w 苷 0**

### ⱍ

^{w}### ⱍ

**u**

**v**

**w**

**a**

*A共1, 1兲 B共3, 2兲* *A共4, 1兲 B共1, 2兲*
*B*共0, 6兲
*A*共2, 1兲

*B*共2, 2兲
*A*共1, 3兲

*B*共4, 2, 1兲
*A*共4, 0, 2兲

*B*共2, 3, 1兲
*A*共0, 3, 1兲

具1, 5典 具3, 1典

具6, 2典 具1, 4典

具0, 0, 6典 具1, 3, 2典

具0, 8, 0典 具3, 0, 1典

### ⱍ

^{a}^{ b}### ⱍ ⱍ

^{a}### ⱍ

**b**苷 具3, 6典
**a**苷 具5, 12典

**b苷 i 2 j**
**a苷 4 i j**

**b苷 2 i j 5k**
**a苷 i 2 j 3k**

**b苷 2 j k**
**a苷 2 i 4 j 4 k**

具4, 2, 4典
**8 i j 4k**

具2, 4, 2典

*x*

**i s3 j** **8 i 6j**

兾3
**v**

### ⱍ

^{v}### ⱍ

^{苷 4}

**v**

*x*

38

40 60 ft兾s

**3i 7j**

**32–33** Find the magnitude of the resultant force and the angle it
makes with the positive -axis.

**32.** **33.**

**34.** *The magnitude of a velocity vector is called speed. Suppose*
that a wind is blowing from the direction N W at a speed of
50 km兾h. (This means that the direction from which the wind
blows is west of the northerly direction.) A pilot is steering
a plane in the direction N E at an airspeed (speed in still air)
of 250 km*兾h. The true course, or track, of the plane is the*
direction of the resul tant of the velocity vectors of the plane
*and the wind. The ground speed of the plane is the magnitude*
of the resultant. Find the true course and the ground speed of
the plane.

**35.** A woman walks due west on the deck of a ship at 5 km兾h. The
ship is moving north at a speed of 35 km兾h. Find the speed and
direction of the woman relative to the surface of the water.

**36.** Ropes 3 m and 5 m in length are fastened to a holiday decora-
tion that is suspended over a town square. The decoration has
a mass of 5 kg. The ropes, fastened at different heights, make
angles of and with the horizontal. Find the tension in
each wire and the magnitude of each tension.

**37.** A clothesline is tied between two poles, 8 m apart. The line
is quite taut and has negligible sag. When a wet shirt with a
mass of 0.8 kg is hung at the middle of the line, the mid point
is pulled down 8 cm. Find the tension in each half of the
clothesline.

**38.** **The tension T at each end of the chain has magnitude 25 N**
(see the figure). What is the weight of the chain?

**39.** A boatman wants to cross a canal that is 3 km wide and wants
to land at a point 2 km upstream from his starting point. The
current in the canal flows at and the speed of his boat
is .

(a) In what direction should he steer?

(b) How long will the trip take?

*x*

300 N

200 N

60° 0 y

x

45 45

60

52 40

3 m 5 m

52° 40°

37° 37°

3.5 km兾h 13 km兾h

20 N

16 N 45° 0 y

x 30°

98845_ch12_ptg01_hr_818-827.qk_98845_ch12_ptg01_hr_818-827 8/18/11 3:26 PM Page 823

**824** **CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE**
**40.** Three forces act on an object. Two of the forces are at an angle

of to each other and have magnitudes 25 N and 12 N. The third is perpendicular to the plane of these two forces and has magnitude 4 N. Calculate the magnitude of the force that would exactly counterbalance these three forces.

**41.** Find the unit vectors that are parallel to the tangent line to the
parabola at the point .

**42.** (a) Find the unit vectors that are parallel to the tangent line to
the curve at the point .

(b) Find the unit vectors that are perpendicular to the tangent line.

(c) Sketch the curve and the vectors in parts (a) and (b), all starting at .

**43.** If , , and are the vertices of a triangle, find
*AB*l

* BC*l

* CA*l
.

**44.** Let be the point on the line segment that is twice as far
from as it is from . If *OA*l

, *OB*l

, and *OC*l
, show
that .

**45.** (a) Draw the vectors , , and
(b) Show, by means of a sketch, that there are scalars and

such that .

(c) Use the sketch to estimate the values of and . (d) Find the exact values of and .

**46.** Suppose that and are nonzero vectors that are not parallel
and is any vector in the plane determined by and . Give
a geometric argument to show that can be written as

for suitable scalars and Then give an argu- ment using components.

**47.** If and , describe the set of all

points such that .

100

共2, 4兲
*y苷 x*^{2}

共兾6, 1兲
*y苷 2 sin x*

*y苷 2 sin x*
共兾6, 1兲
*C*

*B*
*A*

*AB*
*C*

**b**苷
**a**苷

*A*
*B*

**c**苷^{2}3**a**^{1}3**b**

**b**苷 具2, 1典
**a**苷 具3, 2典

**c**苷 具7, 1典.

*t*
*s*
**c****苷 sa tb**

*t*
*s*
*t*

*s*
**b**
**a**

**b**
**a**
**c**

**c**
*t.*

*s*
**c****苷 sa tb**

**r**0*苷 具x*^{0}*, y*0*, z*0典
**r***苷 具x, y, z典*

### ⱍ

^{r}^{ r}^{0}

### ⱍ

^{苷 1}

*共x, y, z兲*

**c**苷

**48.** If , , and , describe the

set of all points such that ,

where .

**49.** Figure 16 gives a geometric demonstration of Property 2 of
vectors. Use components to give an algebraic proof of this
fact for the case .

**50.** Prove Property 5 of vectors algebraically for the case .
Then use similar triangles to give a geometric proof.

**51.** Use vectors to prove that the line joining the midpoints of
two sides of a triangle is parallel to the third side and half
its length.

**52.** Suppose the three coordinate planes are all mirrored and a
light ray given by the vector first strikes the

-plane, as shown in the figure. Use the fact that the angle of incidence equals the angle of reflection to show that the direc- tion of the reflected ray is given by . Deduce that, after being reflected by all three mutually perpendicular mirrors, the resulting ray is parallel to the initial ray. (American space scientists used this principle, together with laser beams and an array of corner mirrors on the moon, to calculate very precisely the distance from the earth to the moon.)

*k*

### ⱍ

^{r}^{1}

^{ r}^{2}

### ⱍ

*n*苷 2

*n*苷 3

**a***苷 具a*^{1}*, a*2*, a*3典
*xz*

**b***苷 具a*^{1}, *a*^{2}*, a*3典

**b**
**a**

z

x

y
**r**2*苷 具x*^{2}*, y*2典
**r**1*苷 具x*^{1}*, y*1典

**r***苷 具x, y典*

### ⱍ

^{r}^{ r}^{1}

### ⱍ

^{}

### ⱍ

^{r}^{ r}^{2}

### ⱍ

^{苷 k}*共x, y兲*

### So far we have added two vectors and multiplied a vector by a scalar. The question arises:

### Is it possible to multiply two vectors so that their product is a useful quantity? One such product is the dot product, whose definition follows. Another is the cross product, which is discussed in the next section.

**Deﬁnition**

### If and **, then the dot product of**

### and is the number given by

### Thus, to find the dot product of and , we multiply corresponding components and add. The result is not a vector. It is a real number, that is, a scalar. For this reason, the dot **product is sometimes called the scalar product (or inner product). Although Definition 1** is given for three-dimensional vectors, the dot product of two-dimensional vectors is defined in a similar fashion:

**a**

*苷 具a*

^{1}

*, a*

2*, a*

3### 典

**b**

*苷 具b*

^{1}

*, b*

2*, b*

3### 典

**a**

**b** **a**

**ⴢ b**

**a**

**ⴢ b 苷 a**

**ⴢ b 苷 a**

^{1}

*b*1

* a*

^{2}

*b*2

* a*

^{3}

*b*3

**a** **b**

*具a*

^{1}

*, a*

2*典 ⴢ 具b*

^{1}

*, b*

2*典 苷 a*

^{1}

*b*1

* a*

^{2}

*b*2

**1**

**12.3** **The Dot Product**

98845_ch12_ptg01_hr_818-827.qk_98845_ch12_ptg01_hr_818-827 8/18/11 3:26 PM Page 824

**830** **CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE**

**1.** Which of the following expressions are meaningful? Which are
meaningless? Explain.

(a) (b)

(c) (d)

(e) (f )

**2–10** Find .

**2.** ,

**3.** ,

**4.** ,

**5.** ,

**6.** ,

**7.** ,

**8.** ,

**9.** , , the angle between and is
**10.** , , the angle between and is

**11–12** **If u is a unit vector, find ** and .

**11.** **12.**

**13.** (a) Show that .

(b) Show that .

**14.** A street vendor sells hamburgers, hot dogs, and soft
drinks on a given day. He charges $2 for a hamburger, $1.50
for a hot dog, and $1 for a soft drink. If and

, what is the meaning of the dot product ?
**15–20** Find the angle between the vectors. (First find an exact
expression and then approximate to the nearest degree.)

**15.** ,

**16.** ,

**17.** ,

**18.** ,

**19.** ,

**20.** ,

**共a ⴢ b兲c**
**共a ⴢ b兲 ⴢ c**

**aⴢ 共b ⫹ c兲**

### ⱍ

^{a}### ⱍ

^{共b ⴢ c兲}### ⱍ

^{a}### ⱍ

^{ⴢ 共b ⫹ c兲}**aⴢ b ⫹ c**
**aⴢ b**

**b** 具0.7, 1.2典
**a** 具⫺2, 3典

**b** 具⫺5, 12典
**a**具⫺2, ^{1}^{3}典

**b** 具2, 5, ⫺1典
**a** 具6, ⫺2, 3典

**b** 具6, ⫺3, ⫺8典
**a**具^{4, 1, }^{1}^{4}典

**b 4i ⫹ 5k**
**a 3i ⫹ 2j ⫺ k**

2兾3
**b**

### ⱍ

^{b}### ⱍ

^{ 5}

**a**

### ⱍ

^{a}### ⱍ

^{ 6}

45⬚
**b**

### ⱍ

^{b}### ⱍ

^{ s6}

**a**

### ⱍ

^{a}### ⱍ

^{ 3}

**uⴢ w**
**uⴢ v**

**w**

**u**

**v**

**w**

**u** **v**

**iⴢ j j ⴢ k k ⴢ i 0**
**iⴢ i j ⴢ j k ⴢ k 1**

*c*
*b*

*a*

**A*** 具a, b, c典*
**Aⴢ P**
**P** 具2, 1.5, 1典

**b** 具2, ⫺1典
**a** 具4, 3典

**b** 具5, 12典
**a** 具⫺2, 5典

**b** 具⫺2, 4, 3典
**a** 具3, ⫺1, 5典

**b** 具2, ⫺1, 0典
**a** 具4, 0, 2典

**b 2i ⫺ k**
**a 4i ⫺ 3j ⫹ k**

**b 4i ⫺ 3k**
**a i ⫹ 2j ⫺ 2k**

**a*** 具s, 2s, 3s典* **b*** 具t, ⫺t, 5t典*
**a i ⫺ 2 j ⫹ 3k** **b 5i ⫹ 9k**

**21–22** Find, correct to the nearest degree, the three angles of the
triangle with the given vertices.

**21.** , ,

**22.** , ,

**23–24** Determine whether the given vectors are orthogonal,
parallel, or neither.

**23.** (a) ,

(b) ,

(c) ,

(d) ,

**24.** (a) ,

(b) ,

(c) ,

**25.** Use vectors to decide whether the triangle with vertices

, , and is right-angled.

**26.** Find the values of such that the angle between the vectors

, and is .

**27.** Find a unit vector that is orthogonal to both and .
**28.** Find two unit vectors that make an angle of with

.

**29–30** Find the acute angle between the lines.

**29.** ,

**30.** ,

**31–32** Find the acute angles between the curves at their points of
intersection. (The angle between two curves is the angle between
their tangent lines at the point of intersection.)

**31.** ,

**32.** , ,

**33–37** Find the direction cosines and direction angles of the vector.

(Give the direction angles correct to the nearest degree.)

**33.** **34.**

**35.** **36.**

**37.** , where

**38.** If a vector has direction angles and , find the
third direction angle .

*P共2, 0兲 Q共0, 3兲 R共3, 4兲*

*A共1, 0, ⫺1兲 B共3, ⫺2, 0兲 C共1, 3, 3兲*

**a** 具⫺5, 3, 7典 **b** 具6, ⫺8, 2典
**a** 具4, 6典 **b** 具⫺3, 2典

**a ⫺i ⫹ 2 j ⫹ 5k b 3i ⫹ 4 j ⫺ k**
**a 2i ⫹ 6 j ⫺ 4k b ⫺3i ⫺ 9 j ⫹ 6k**
**u** 具⫺3, 9, 6典 **v** 具4, ⫺12, ⫺8典
**u i ⫺ j ⫹ 2k v 2i ⫺ j ⫹ k**
**u*** 具a, b, c典* **v*** 具⫺b, a, 0典*

*P共1, ⫺3, ⫺2兲 Q共2, 0, ⫺4兲* *R*共6, ⫺2, ⫺5兲
*x*

具2, 1, ⫺1典 *具1, x, 0典* 45⬚

**i⫹ j** **i⫹ k**
60⬚

**v** 具3, 4典

*2x⫺ y 3 3x ⫹ y 7*
*x⫹ 2y 7 5x ⫺ y 2*

*y x*^{2} *y x*^{3}

*y sin x y cos x 0 艋 x 艋*兾2

具2, 1, 2典 具6, 3, ⫺2典

**i⫺ 2j ⫺ 3k** ^{1}2**i⫹ j ⫹ k**

*具c, c, c典* *c*⬎ 0

␣ 兾4  兾3

␥

**12.3** **Exercises**

**1.** Homework Hints available at stewartcalculus.com

98845_ch12_ptg01_hr_828-837.qk_98845_ch12_ptg01_hr_828-837 8/18/11 3:26 PM Page 830

**SECTION 12.3 THE DOT PRODUCT** **831**
**39–44** Find the scalar and vector projections of onto .

**39.** ,

**40.** ,

**41.** ,

**42.** ,

**43.** ,

**44.** ,

**45.** Show that the vector is orthogonal to .
**(It is called an orthogonal projection of .)**

**46.** For the vectors in Exercise 40, find and illustrate by
drawing the vectors , , , and .

**47.** If , find a vector such that .

**48.** Suppose that and are nonzero vectors.

(a) Under what circumstances is ? (b) Under what circumstances is ?

**49.** Find the work done by a force that moves
an object from the point to the point along
a straight line. The distance is measured in meters and the force
in newtons.

**50.** A tow truck drags a stalled car along a road. The chain makes
an angle of with the road and the tension in the chain is
1500 N. How much work is done by the truck in pulling the
car 1 km?

**51.** A woman exerts a horizontal force of 140 N on a crate as
she pushes it up a ramp that is 4 m long and inclined at an
angle of above the horizontal. Find the work done on
the box.

**52.** Find the work done by a force of 100 N acting in the direction
N W in moving an object 5 m due west.

**53.** Use a scalar projection to show that the distance from a point

to the line is

Use this formula to find the distance from the point to

the line .

**54.** If , and , show

that the vector equation represents a sphere, and find its center and radius.

**b** **a**

**a** 具⫺5, 12典 **b** 具4, 6典
**a** 具1, 4典 **b** 具2, 3典

**b** 具1, 2, 3典
**a** 具3, 6, ⫺2典

**b** 具5, ⫺1, 4典
**a** 具⫺2, 3, ⫺6典

**b j ⫹**^{1}2**k**
**a 2i ⫺ j ⫹ 4k**

**b i ⫺ j ⫹ k**
**a i ⫹ j ⫹ k**

**a**
orth_{a}**b b ⫺ proj****a****b**

**b**
orth_{a}**b**

orth_{a}**b**
proj_{a}**b**

**b**
**a**

comp_{a}**b** 2
**b**

**a** 具3, 0, ⫺1典
**b**
**a**

comp_{a}**b** comp**b****a**
proj_{a}**b** proj**b****a**
**F 8 i ⫺ 6 j ⫹ 9k**

共6, 12, 20兲 共0, 10, 8兲

30⬚

*ax⫹ by ⫹ c 0*
*P*1*共x*^{1}*, y*1兲

### ⱍ

^{a x}^{1}

^{⫹ by}^{1}

^{⫹ c}### ⱍ

sa^{2}*⫹ b*^{2 }

共⫺2, 3兲
*3x⫺ 4y ⫹ 5 0*

**b*** 具b*^{1}*, b*2*, b*3典
**r**** 具x, y, z典, a 具a**^{1}*, a*2*, a*3典

**共r ⫺ a兲 ⴢ 共r ⫺ b兲 0**
20⬚

50⬚

**55.** Find the angle between a diagonal of a cube and one of its
edges.

**56.** Find the angle between a diagonal of a cube and a diagonal of
one of its faces.

**57.** A molecule of methane, , is structured with the four hydro-
gen atoms at the vertices of a regular tetrahedron and the car-
*bon atom at the centroid. The bond angle is the angle formed*
by the H— C —H combination; it is the angle between the
lines that join the carbon atom to two of the hydrogen atoms.

Show that the bond angle is about *. Hint: Take the *
vertices of the tetrahedron to be the points , ,

, and , as shown in the figure. Then the centroid is .

**58.** If , where , , and are all nonzero vectors,
show that bisects the angle between and .

**59.** Prove Properties 2, 4, and 5 of the dot product (Theorem 2).

**60.** Suppose that all sides of a quadrilateral are equal in length and
opposite sides are parallel. Use vector methods to show that the
diagonals are perpendicular.

**61.** Use Theorem 3 to prove the Cauchy-Schwarz Inequality:

**62.** The Triangle Inequality for vectors is

(a) Give a geometric interpretation of the Triangle Inequality.

(b) Use the Cauchy-Schwarz Inequality from Exercise 61 to
*prove the Triangle Inequality. [Hint: Use the fact that*

and use Property 3 of the dot product.]

**63.** The Parallelogram Law states that

(a) Give a geometric interpretation of the Parallelogram Law.

(b) Prove the Parallelogram Law. (See the hint in Exercise 62.)
**64.** Show that if and are orthogonal, then the vectors

and must have the same length.

CH4

109.5⬚ [

共1, 0, 0兲 共0, 1, 0兲 共0, 0, 1兲 共1, 1, 1兲

(^{1}2, ^{1}_{2}, ^{1}_{2}) ]

H

H H

H C

x

y z

**c**

### ⱍ

^{a}### ⱍ

^{b}^{⫹}

### ⱍ

^{b}### ⱍ

^{a}

^{a b}

^{c}**c** **a** **b**

### ⱍ

^{a}^{ⴢ b}### ⱍ

^{艋}

### ⱍ

^{a}### ⱍⱍ

^{b}### ⱍ ⱍ

^{a}^{⫹ b}### ⱍ

^{艋}

### ⱍ

^{a}### ⱍ

^{⫹}

### ⱍ

^{b}### ⱍ

### ⱍ

^{a}^{⫹ b}### ⱍ

^{2}

**共a ⫹ b兲 ⭈ 共a ⫹ b兲**

### ⱍ

^{a}^{⫹ b}### ⱍ

^{2}

^{⫹}

### ⱍ

^{a}^{⫺ b}### ⱍ

^{2}

^{ 2}

### ⱍ

^{a}### ⱍ

^{2}

^{⫹ 2}

### ⱍ

^{b}### ⱍ

^{2}

**u⫹ v** **u⫺ v**

**u** **v**

98845_ch12_ptg01_hr_828-837.qk_98845_ch12_ptg01_hr_828-837 8/18/11 3:26 PM Page 831

**838** **CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE**

### where is the angle between the position and force vectors. Observe that the only com- ponent of that can cause a rotation is the one perpendicular to , that is, . The magnitude of the torque is equal to the area of the parallelogram determined by and . A bolt is tightened by applying a 40-N force to a 0.25-m wrench as shown in Figure 5. Find the magnitude of the torque about the center of the bolt.

SOLUTION

### The magnitude of the torque vector is

### If the bolt is right-threaded, then the torque vector itself is

### where is a unit vector directed down into the page.

###

**F** **r**

### ⱍ

^{F}### ⱍ ^{sin } ^{}

**r** **F**

**EXAMPLE 6**

### ⱍ ^{} ⱍ ^{苷} ⱍ

^{r}^{⫻ F} ⱍ ^{苷} ⱍ

^{⫻ F}

^{r}### ⱍⱍ

^{F}### ⱍ ^{sin 75} ⬚ 苷 共0.25兲共40兲 sin 75⬚

### 苷 10 sin 75⬚ ⬇ 9.66 N⭈m

### 苷 ⱍ ^{} ⱍ

^{n}^{⬇ 9.66 n}

^{⬇ 9.66 n}

**FIGURE 5** **n**

75° 0.25 m 40 N

**1–7** Find the cross product and verify that it is orthogonal to
**both a and b.**

**1.** ,

**2.** ,

**3.** ,

**4.** ,

**5.** ,

**6.** ,

**7.** ,

**8.** **If a苷 i ⫺ 2k and b 苷 j ⫹ k, find a ⫻ b. Sketch a, b, and **
**a⫻ b as vectors starting at the origin.**

**9–12** Find the vector, not with determinants, but by using proper-
ties of cross products.

**9.** **10.**

**11.** **12.**

**13.** State whether each expression is meaningful. If not, explain
why. If so, state whether it is a vector or a scalar.

(a) (b)

(c) (d)

(e) (f )

**a⫻ b**

**a**苷 具6, 0, ⫺2典 **b**苷 具0, 8, 0典
**a**苷 具1, 1, ⫺1典 **b**苷 具2, 4, 6典
**a苷 i ⫹ 3j ⫺ 2k b 苷 ⫺i ⫹ 5k**
**a苷 j ⫹ 7k b 苷 2i ⫺ j ⫹ 4k**
**a苷 i ⫺ j ⫺ k b 苷**^{1}2**i⫹ j ⫹**^{1}2**k**

**a****苷 ti ⫹ cos tj ⫹ sin tk b 苷 i ⫺ sin tj ⫹ cos tk****a****苷 具t, 1, 1兾t典 b 苷 具t**^{2}*, t*^{2}, 1典

**共i ⫻ j兲 ⫻ k** **k⫻ 共i ⫺ 2j兲**

**共 j ⫺ k兲 ⫻ 共k ⫺ i兲** **共i ⫹ j兲 ⫻ 共i ⫺ j兲**

**aⴢ 共b ⫻ c兲** **a⫻ 共b ⴢ c兲**

**a⫻ 共b ⫻ c兲** **aⴢ 共b ⴢ c兲**

**共a ⴢ b兲 ⫻ 共c ⴢ d兲** **共a ⫻ b兲 ⴢ 共c ⫻ d兲**

**14–15** Find **and determine whether u⫻ v is directed into**
the page or out of the page.

**14.** **15.**

**16.** The figure shows a vector in the -plane and a vector in
the direction of . Their lengths are and

(a) Find .

(b) Use the right-hand rule to decide whether the com ponents of are positive, negative, or 0.

**17.** If and , find and .

**18.** If , , and , show that

.

**19.** Find two unit vectors orthogonal to both and
.

### ⱍ

^{u}^{⫻ v}### ⱍ

45°

**| u |=4**

**| v |=5** **| v |=16**

120°

**| u |=12**

**a** *xy* **b**

**k**

### ⱍ

^{a}### ⱍ

^{苷 3}

### ⱍ

^{b}### ⱍ

^{苷 2.}

### ⱍ

^{a}^{⫻ b}### ⱍ

**a⫻ b**

x

z

y
**b**

**a**

**a**苷 具2, ⫺1, 3典 **b**苷 具4, 2, 1典 **a⫻ b** **b⫻ a**
**a苷 具1, 0, 1典 b 苷 具2, 1, ⫺1典** **c**苷 具0, 1, 3典
**a⫻ 共b ⫻ c兲 苷 共a ⫻ b兲 ⫻ c**

具3, 2, 1典 具⫺1, 1, 0典

**12.4** **Exercises**

**1.** Homework Hints available at stewartcalculus.com

98845_ch12_ptg01_hr_838-847.qk_98845_ch12_ptg01_hr_838-847 8/18/11 3:27 PM Page 838

**SECTION 12.4 THE CROSS PRODUCT** **839**
**20.** Find two unit vectors orthogonal to both and .

**21.** Show that for any vector in .
**22.** Show that for all vectors and in .
**23.** Prove Property 1 of Theorem 11.

**24.** Prove Property 2 of Theorem 11.

**25.** Prove Property 3 of Theorem 11.

**26.** Prove Property 4 of Theorem 11.

**27.** Find the area of the parallelogram with vertices ,

, , and .

**28.** Find the area of the parallelogram with vertices ,

, , and .

**29–32** (a) Find a nonzero vector orthogonal to the plane through
the points , , and , and (b) find the area of triangle .

**29.** , ,

**30.** , ,

**31.** , ,

**32.** , ,

**33–34** Find the volume of the parallelepiped determined by the
vectors , , and .

**33.** , ,

**34.** , ,

**35–36** Find the volume of the parallelepiped with adjacent edges
, , and .

**35.** , , ,

**36.** , , ,

**37.** Use the scalar triple product to verify that the vectors

, , and

are coplanar.

**38.** Use the scalar triple product to determine whether the points

, , , and lie in the

same plane.

**39.** A bicycle pedal is pushed by a foot with a 60-N force as
shown. The shaft of the pedal is 18 cm long. Find the
magnitude of the torque about .

**共a ⫻ b兲 ⴢ b 苷 0** **a** **b** *V*3

**j⫺ k** **i⫹ j**
*V*3

**a**
**0⫻ a 苷 0 苷 a ⫻ 0**

*A*共⫺2, 1兲
*D*共2, ⫺1兲

*C*共4, 2兲
*B*共0, 4兲

*K*共1, 2, 3兲
*N*共3, 7, 3兲

*M*共3, 8, 6兲
*L*共1, 3, 6兲

*PQR*
*R*

*Q*
*P*

*R*共5, 3, 1兲
*Q*共4, 1, ⫺2兲

*P*共0, ⫺2, 0兲

*R*共4, 3, ⫺1兲
*Q*共0, 5, 2兲

*P*共⫺1, 3, 1兲

*P共1, 0, 1兲 Q共⫺2, 1, 3兲 R共4, 2, 5兲*
*P共0, 0, ⫺3兲 Q共4, 2, 0兲 R共3, 3, 1兲*

**c**
**b**
**a**

*PS*
*PR*
*PQ*

*S*共3, 6, 1兲
*R*共1, 4, ⫺1兲

*Q*共2, 3, 2兲
*P*共⫺2, 1, 0兲

*S*共0, 4, 2兲
*R*共5, 1, ⫺1兲

*Q*共⫺1, 2, 5兲
*P*共3, 0, 1兲

**w苷 5i ⫹ 9 j ⫺ 4 k**
**v苷 3i ⫺ j**

**u苷 i ⫹ 5 j ⫺ 2 k**

*D*共3, 6, ⫺4兲
*C*共5, 2, 0兲

*B*共3, ⫺1, 6兲
*A*共1, 3, 2兲

*P*

10° 70° 60 N

P
**a**苷 具6, 3, ⫺1典 **b**苷 具0, 1, 2典 **c**苷 具4, ⫺2, 5典
**a苷 i ⫹ j ⫺ k b 苷 i ⫺ j ⫹ k c 苷 ⫺i ⫹ j ⫹ k**

**40.** Find the magnitude of the torque about if a 240-N force is
applied as shown.

**41.** A wrench 30 cm long lies along the positive -axis and grips a
bolt at the origin. A force is applied in the direction

at the end of the wrench. Find the magnitude of the force needed to supply of torque to the bolt.

**42.** **Let v苷 5j and let u be a vector with length 3 that starts at **
the origin and rotates in the -plane. Find the maximum and
**minimum values of the length of the vector u⫻ v. In what**
**direction does u⫻ v point?**

**43.** If and , find the angle between

and .

**44.** (a) Find all vectors such that

(b) Explain why there is no vector such that

**45.** (a) Let be a point not on the line that passes through the
points and . Show that the distance from the point
to the line is

where *QR*l

and *QP*l
.

(b) Use the formula in part (a) to find the distance from the point to the line through and

.

**46.** (a) Let be a point not on the plane that passes through the
points , , and . Show that the distance from to the
plane is

where *QR*l
, *QS*l

, and *QP*l
.

(b) Use the formula in part (a) to find the distance from the point to the plane through the points ,

, and .

**47.** Show that .

**48.** If , show that

*y*

具0, 3, ⫺4典 100 N⭈m

*xy*

**aⴢ b 苷 s3** **a⫻ b 苷 具1, 2, 2典** **a**

**b**

**v**

**具1, 2, 1典 ⫻ v 苷 具3, 1, ⫺5典**
**v**
**具1, 2, 1典 ⫻ v 苷 具3, 1, 5典**

*P* *L*

*Q* *R* *d* *P*

*L*

*d*苷

### ⱍ

^{a}^{⫻ b}### ⱍ ⱍ

^{a}### ⱍ

**a**苷 **b**苷

*P*共1, 1, 1兲 *Q*共0, 6, 8兲

*R*共⫺1, 4, 7兲
*P*

*Q R* *S* *d* *P*

*d*苷

### ⱍ

^{a}^{ⴢ 共b ⫻ c兲}### ⱍ ⱍ

^{a}^{⫻ b}### ⱍ

**a**苷 **b**苷 **c**苷

*P*共2, 1, 4兲 *Q*共1, 0, 0兲

*R*共0, 2, 0兲 *S*共0, 0, 3兲

### ⱍ

^{a}^{⫻ b}### ⱍ

^{2}

^{苷}

### ⱍ

^{a}### ⱍ

^{2}

### ⱍ

^{b}### ⱍ

^{2}

^{⫺ 共a ⴢ b兲}^{2}

**a⫻ b 苷 b ⫻ c 苷 c ⫻ a**
**a⫹ b ⫹ c 苷 0**

*P*

30° 240 N

2 m 2 m

P 98845_ch12_ptg01_hr_838-847.qk_98845_ch12_ptg01_hr_838-847 8/18/11 3:27 PM Page 839

**840** **CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE**

**49.** Prove that .

**50.** Prove Property 6 of Theorem 11, that is,

**51.** Use Exercise 50 to prove that

**52.** Prove that

**53.** Suppose that .

(a) If , does it follow that ? (b) If , does it follow that ?

(c) If and , does it follow

that ?
**a苷 0**

**b苷 c**
**aⴢ b 苷 a ⴢ c**

**b苷 c**
**a⫻ b 苷 a ⫻ c**

**aⴢ b 苷 a ⴢ c** **a⫻ b 苷 a ⫻ c**
**b苷 c**

**共a ⫺ b兲 ⫻ 共a ⫹ b兲 苷 2共a ⫻ b兲**

**a⫻ 共b ⫻ c兲 苷 共a ⴢ c兲b ⫺ 共a ⴢ b兲c**

**a⫻ 共b ⫻ c兲 ⫹ b ⫻ 共c ⫻ a兲 ⫹ c ⫻ 共a ⫻ b兲 苷 0**

**共a ⫻ b兲 ⴢ 共c ⫻ d兲 苷**

## 冟

^{a}^{a}^{ⴢ c}^{ⴢ d}

^{b}^{b}^{ⴢ c}^{ⴢ d}## 冟

**54.** If , , and are noncoplanar vectors, let

(These vectors occur in the study of crystallography. Vectors
of the form , where each is an integer,
*form a lattice for a crystal. Vectors written similarly in terms of*

, , and *form the reciprocal lattice.)*
(a) Show that is perpendicular to if .

(b) Show that for .

(c) Show that .

**k**1 **k**2 **k**3

**k***i* **v***j* *i苷 j*

**k***i***ⴢ v*** ^{i}*苷 1

*i*苷 1, 2, 3

**k**1

**ⴢ 共k**

^{2}

**⫻ k**

^{3}兲 苷 1

**v**1**ⴢ 共v**^{2}**⫻ v**^{3}兲
**v**3

**v**2

**v**1

**k**2苷 **v**3**⫻ v**^{1}
**v**1**ⴢ 共v**^{2}**⫻ v**^{3}兲
**k**1苷 **v**2**⫻ v**^{3}

**v**1**ⴢ 共v**^{2}**⫻ v**^{3}兲

**k**3苷 **v**1**⫻ v**^{2}
**v**1**ⴢ 共v**^{2}**⫻ v**^{3}兲

*n**i*

*n*1**v**1*⫹ n*^{2}**v**2*⫹ n*^{3}**v**3

**D I S C O V E RY P R O J E C T** **THE GEOMETRY OF A TETRAHEDRON**

A tetrahedron is a solid with four vertices, , , , and , and four triangular faces, as shown in the figure.

**1.** Let , , , and be vectors with lengths equal to the areas of the faces opposite the
vertices , , , and , respectively, and directions perpendicular to the respective faces and
pointing outward. Show that

**2.** The volume of a tetrahedron is one-third the distance from a vertex to the opposite face,
times the area of that face.

(a) Find a formula for the volume of a tetrahedron in terms of the coordinates of its vertices , , , and .

(b) Find the volume of the tetrahedron whose vertices are , , , and .

**3.** *Suppose the tetrahedron in the figure has a trirectangular vertex S. (This means that the*
*three angles at S are all right angles.) Let A, B, and C be the areas of the three faces that*
*meet at S, and let D be the area of the opposite face PQR. Using the result of Problem 1, *
or otherwise, show that

(This is a three-dimensional version of the Pythagorean Theorem.)

*P Q R* *S*

**v**1 **v**2 **v**3 **v**4

*P Q R* *S*

**v**1**⫹ v**^{2}**⫹ v**^{3}**⫹ v**^{4}**苷 0**
*V*

*P Q R* *S*

*P共1, 1, 1兲 Q共1, 2, 3兲 R共1, 1, 2兲*
*S*共3, ⫺1, 2兲

*D*^{2}*苷 A*^{2}*⫹ B*^{2}*⫹ C*^{2}
P

Q R

S

### A line in the -plane is determined when a point on the line and the direction of the line (its slope or angle of inclination) are given. The equation of the line can then be written using the point-slope form.

### Likewise, a line in three-dimensional space is determined when we know a point on and the direction of . In three dimensions the direction of a line is con- veniently described by a vector, so we let be a vector parallel to . Let be an arbi- trary point on and let and be the position vectors of and (that is, they have

*xy*

*L*

*P*0

*共x*

^{0}

*, y*

0*, z*

0### 兲

*L*

*L*

*P*

*共x, y, z兲*

*L*

**v**

*P*
*P*0

**r**
**r**0

*L*

**12.5** **Equations of Lines and Planes**

98845_ch12_ptg01_hr_838-847.qk_98845_ch12_ptg01_hr_838-847 8/18/11 3:27 PM Page 840

**848** **CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE**

**1.** Determine whether each statement is true or false.

(a) Two lines parallel to a third line are parallel.

(b) Two lines perpendicular to a third line are parallel.

(c) Two planes parallel to a third plane are parallel.

(d) Two planes perpendicular to a third plane are parallel.

(e) Two lines parallel to a plane are parallel.

(f ) Two lines perpendicular to a plane are parallel.

(g) Two planes parallel to a line are parallel.

(h) Two planes perpendicular to a line are parallel.

( i) Two planes either intersect or are parallel.

( j) Two lines either intersect or are parallel.

(k) A plane and a line either intersect or are parallel.

**2–5** Find a vector equation and parametric equations for the line.

**2.** The line through the point and parallel to the
vector

**3.** The line through the point and parallel to the
vector

**4.** The line through the point and parallel to the line
, ,

**5.** The line through the point (1, 0, 6) and perpendicular to the
plane

**6–12** Find parametric equations and symmetric equations for the
line.

**6.** The line through the origin and the point
**7.** The line through the points and
**8.** The line through the points and
**9.** The line through the points and

**10.** The line through and perpendicular to both
and

**11.** The line through and parallel to the line

**12.** The line of intersection of the planes
and

**13.** Is the line through and parallel to the

line through and ?

**14.** Is the line through and perpendicular to the

line through and ?

**15.** (a) Find symmetric equations for the line that passes
through the point and is parallel to the vector

.

(b) Find the points in which the required line in part (a) inter- sects the coordinate planes.

共6, ⫺5, 2兲

具^{1, 3, }^{⫺}^{2}^{3}典

共2, 2.4, 3.5兲
**3 i⫹ 2j ⫺ k**

共0, 14, ⫺10兲
*z苷 3 ⫹ 9t*
*y苷 6 ⫺ 3t*

*x苷 ⫺1 ⫹ 2t*

*x⫹ 3y ⫹ z 苷 5*

共2, 1, ⫺3兲

(0, ^{1}2, 1)

共1.0, 2.4, 4.6兲 共2.6, 1.2, 0.3兲 共⫺8, 1, 4兲 共3, ⫺2, 4兲

**i⫹ j**
共2, 1, 0兲

**j⫹ k**

共1, ⫺1, 1兲
*x*⫹ 2 苷^{1}2*y苷 z ⫺ 3*

*x⫹ 2y ⫹ 3z 苷 1*
*x⫺ y ⫹ z 苷 1*

共⫺2, 0, ⫺3兲 共⫺4, ⫺6, 1兲

共5, 3, 14兲 共10, 18, 4兲

共1, 1, 1兲 共⫺2, 4, 0兲

共3, ⫺1, ⫺8兲 共2, 3, 4兲

共1, ⫺5, 6兲 具⫺1, 2, ⫺3典

共1, 2, 3兲

**16.** (a) Find parametric equations for the line through that
is perpendicular to the plane .

(b) In what points does this line intersect the coordinate planes?

**17.** Find a vector equation for the line segment from
to .

**18.** Find parametric equations for the line segment from
to .

**19–22** Determine whether the lines and are parallel, skew, or
intersecting. If they intersect, find the point of intersection.

**19.** : , ,

: , ,

**20.** : , ,

: , ,

**21.** :

:

**22.** :

:

**23–40** Find an equation of the plane.

**23.** The plane through the point and perpendicular to the
vector

**24.** The plane through the point and with normal
vector

**25.** The plane through the point and with normal
vector

**26.** The plane through the point and perpendicular to the

line , ,

**27.** The plane through the point and parallel to the
plane

**28.** The plane through the point and parallel to the plane

**29.** The plane through the point and parallel to the plane

**30.** The plane that contains the line , ,
and is parallel to the plane

**31.** The plane through the points , , and
**32.** The plane through the origin and the points

and

共2, ⫺1, 4兲 共4, 6, 1兲

共10, 3, 1兲 共5, 6, ⫺3兲

共2, 4, 6兲
*x⫺ y ⫹ 3z 苷 7*

*L*1 *L*2

*L*1 *x苷 3 ⫹ 2t y 苷 4 ⫺ t z 苷 1 ⫹ 3t*
*L*2 *x苷 1 ⫹ 4s y 苷 3 ⫺ 2s z 苷 4 ⫹ 5s*
*L*1 *x苷 5 ⫺ 12t y 苷 3 ⫹ 9t*

*L*2 *x苷 3 ⫹ 8s y 苷 ⫺6s z 苷 7 ⫹ 2s*
*L*1

*x*⫺ 2

1 苷 *y*⫺ 3

⫺2 苷 *z*⫺ 1

⫺3
*L*2

*x*⫺ 3

1 苷 *y*⫹ 4

3 苷 *z*⫺ 2

⫺7
*L*1

*x*

1 苷 *y*⫺ 1

⫺1 苷 *z*⫺ 2
3
*L*2

*x*⫺ 2

2 苷 *y*⫺ 3

⫺2 苷 *z*
7

(⫺1, ^{1}2, 3)

**i⫹ 4j ⫹ k**

共2, 0, 1兲
*x苷 3t y 苷 2 ⫺ t z 苷 3 ⫹ 4t*

共1, ⫺1, ⫺1兲
*5x⫺ y ⫺ z 苷 6*

共2, 4, 6兲
*z苷 x ⫹ y*

(1, ^{1}2, ^{1}3)

*x⫹ y ⫹ z 苷 0*

*x苷 1 ⫹ t y 苷 2 ⫺ t*

*z苷 4 ⫺ 3t* *5x⫹ 2y ⫹ z 苷 1*

共0, 1, 1兲 共1, 0, 1兲 共1, 1, 0兲 共2, ⫺4, 6兲 共5, 1, 3兲

*z苷 1 ⫺ 3t*

共6, 3, 2兲 具⫺2, 1, 5典

共4, 0, ⫺3兲
**j⫹ 2k**

**12.5** **Exercises**

**1.** Homework Hints available at stewartcalculus.com

98845_ch12_ptg01_hr_848-857.qk_98845_ch12_ptg01_hr_848-857 8/18/11 3:28 PM Page 848

**SECTION 12.5 EQUATIONS OF LINES AND PLANES** **849**
**33.** The plane through the points , , and

**34.** The plane that passes through the point and contains

the line , ,

**35.** The plane that passes through the point and contains

the line , ,

**36.** The plane that passes through the point and
contains the line with symmetric equations

**37.** The plane that passes through the point and contains
the line of intersection of the planes and

**38.** The plane that passes through the points and
and is perpendicular to the plane

**39.** The plane that passes through the point and is perpen-
dicular to the planes and

**40.** The plane that passes through the line of intersection of the
planes and and is perpendicular to the
plane

**41–44** Use intercepts to help sketch the plane.

**41.** **42.**

**43.** **44.**

**45–47** Find the point at which the line intersects the given plane.

**45.** , , ;

**46.** , , ;

**47.** ;

**48.** Where does the line through and intersect

the plane ?

**49.** Find direction numbers for the line of intersection of the planes
and .

**50.** Find the cosine of the angle between the planes
and .

**51–56** Determine whether the planes are parallel, perpendicular, or
neither. If neither, find the angle between them.

**51.** ,

**52.** ,

**53.** ,

**54.** ,

**55.** ,

**56.** ,

共⫺1, ⫺2, ⫺3兲

共1, 2, 3兲
*x苷 3t y 苷 1 ⫹ t z 苷 2 ⫺ t*

共6, 0, ⫺2兲
*z苷 7 ⫹ 4t*
*y苷 3 ⫹ 5t*

*x苷 4 ⫺ 2t*

共1, ⫺1, 1兲
*x苷 2y 苷 3z*
共⫺1, 2, 1兲
*x⫹ y ⫺ z 苷 2*
*2 x⫺ y ⫹ 3z 苷 1*

共0, ⫺2, 5兲
*2z苷 5x ⫹ 4y*
共⫺1, 3, 1兲

共1, 5, 1兲
*x⫹ 3z 苷 4*
*2x⫹ y ⫺ 2z 苷 2*

*y⫹ 2z 苷 3*
*x⫺ z 苷 1*

*x⫹ y ⫺ 2z 苷 1*

*3x⫹ y ⫹ 2z 苷 6*
*2x⫹ 5y ⫹ z 苷 10*

*6x⫹ 5y ⫺ 3z 苷 15*
*6x⫺ 3y ⫹ 4z 苷 6*

*x⫺ y ⫹ 2z 苷 9*
*z苷 5t*

*y苷 2 ⫹ t*
*x苷 3 ⫺ t*

*x⫹ 2y ⫺ z ⫹ 1 苷 0*
*z苷 2 ⫺ 3t*

*y苷 4t*
*x苷 1 ⫹ 2t*

*4x⫺ y ⫹ 3z 苷 8*
*x苷 y ⫺ 1 苷 2z*

共4, ⫺2, 2兲 共1, 0, 1兲

*x⫹ y ⫹ z 苷 6*

*x⫹ z 苷 0*
*x⫹ y ⫹ z 苷 1*

*x⫹ y ⫹ z 苷 0*
*x⫹ 2y ⫹ 3z 苷 1*

*⫺3x ⫹ 6y ⫹ 7z 苷 0*
*x⫹ 4y ⫺ 3z 苷 1*

*3x⫺ 12y ⫹ 6z 苷 1*
*2z苷 4y ⫺ x*

*x⫺ y ⫹ z 苷 1*
*x⫹ y ⫹ z 苷 1*

*x⫹ 6y ⫹ 4z 苷 3*
*2 x⫺ 3y ⫹ 4z 苷 5*

*8y苷 1 ⫹ 2x ⫹ 4z*
*x苷 4y ⫺ 2z*

*2 x⫺ y ⫹ 2z 苷 1*
*x⫹ 2y ⫹ 2z 苷 1*

共8, 2, 4兲

共3, ⫺1, 2兲 **57–58** (a) Find parametric equations for the line of intersection of
the planes and (b) find the angle between the planes.

**57.** ,

**58.** ,

**59–60** Find symmetric equations for the line of intersection of the
planes.

**59.** ,

**60.** ,

**61.** Find an equation for the plane consisting of all points that are
equidistant from the points and .

**62.** Find an equation for the plane consisting of all points that are
equidistant from the points and .

**63.** Find an equation of the plane with -intercept , -intercept ,
and -intercept .

**64.** (a) Find the point at which the given lines intersect:

(b) Find an equation of the plane that contains these lines.

**65.** Find parametric equations for the line through the point
that is parallel to the plane and
perpendicular to the line , , .
**66.** Find parametric equations for the line through the point

that is perpendicular to the line , , and intersects this line.

**67.** Which of the following four planes are parallel? Are any of
them identical?

**68.** Which of the following four lines are parallel? Are any of them
identical?

, ,

, ,

**69–70** Use the formula in Exercise 45 in Section 12.4 to find the
distance from the point to the given line.

**69.** ; , ,

**70.** ; , ,

*3x⫺ 2y ⫹ z 苷 1 2x ⫹ y ⫺ 3z 苷 3*

*5x⫺ 2y ⫺ 2z 苷 1 4x ⫹ y ⫹ z 苷 6*
*z苷 2x ⫺ y ⫺ 5 z 苷 4x ⫹ 3y ⫺ 5*

共1, 0, ⫺2兲 共3, 4, 0兲

共2, 5, 5兲 共⫺6, 3, 1兲

*x* *a y* *b*

*z* *c*

**r***苷 具1, 1, 0典 ⫹ t 具1, ⫺1, 2典*
**r***苷 具2, 0, 2典 ⫹ s具⫺1, 1, 0典*

共0, 1, 2兲 *x⫹ y ⫹ z 苷 2*

*x苷 1 ⫹ t y 苷 1 ⫺ t z 苷 2t*

共0, 1, 2兲 *x苷 1 ⫹ t*

*y苷 1 ⫺ t z 苷 2t*

*P*1*: 3x⫹ 6y ⫺ 3z 苷 6* *P*2*: 4x⫺ 12y ⫹ 8z 苷 5*
*P*3*: 9y苷 1 ⫹ 3x ⫹ 6z* *P*4*: z苷 x ⫹ 2y ⫺ 2*

*L*1*: x苷 1 ⫹ 6t y 苷 1 ⫺ 3t z 苷 12t ⫹ 5*
*L*2*: x苷 1 ⫹ 2t y 苷 t z 苷 1 ⫹ 4t*
*L*3*: 2x⫺ 2 苷 4 ⫺ 4y 苷 z ⫹ 1*
*L*4**: r***苷 具3, 1, 5典 ⫹ t 具4, 2, 8典*

*共4, 1, ⫺2兲 x 苷 1 ⫹ t y 苷 3 ⫺ 2t z 苷 4 ⫺ 3t*
*共0, 1, 3兲 x 苷 2t y 苷 6 ⫺ 2t z 苷 3 ⫹ t*

*x⫹ 2y ⫹ 2z 苷 1*
*x⫹ y ⫹ z 苷 1*

98845_ch12_ptg01_hr_848-857.qk_98845_ch12_ptg01_hr_848-857 8/18/11 3:28 PM Page 849

**850** **CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE**
**71–72** Find the distance from the point to the given plane.

**71.** ,

**72.** ,

**73–74** Find the distance between the given parallel planes.

**73.** ,

**74.** ,

**75.** Show that the distance between the parallel planes
and is

**76.** Find equations of the planes that are parallel to the plane
and two units away from it.

**77.** Show that the lines with symmetric equations and
are skew, and find the distance between
these lines.

*2x⫺ 3y ⫹ z 苷 4 4x ⫺ 6y ⫹ 2z 苷 3*
*6z苷 4y ⫺ 2x 9z 苷 1 ⫺ 3x ⫹ 6y*

*ax⫹ by ⫹ cz ⫹ d*^{1}苷 0 *ax⫹ by ⫹ cz ⫹ d*^{2}苷 0

*D*苷

### ⱍ

^{d}^{1}

^{⫺ d}^{2}

### ⱍ

sa^{2}*⫹ b*^{2}*⫹ c*^{2 }

*x⫹ 2y ⫺ 2z 苷 1*

*x苷 y 苷 z*
*x⫹ 1 苷 y兾2 苷 z兾3*

*3x⫹ 2y ⫹ 6z 苷 5*
共1, ⫺2, 4兲

*x⫺ 2y ⫺ 4z 苷 8*
共⫺6, 3, 5兲

**78.** Find the distance between the skew lines with parametric

equations , , , and ,

, .

**79.** Let be the line through the origin and the point .
Let be the line through the points and .
Find the distance between and .

**80.** Let be the line through the points and .
Let be the line of intersection of the planes and ,

where is the plane and is the plane

through the points , , and . Calculate the distance between and .

**81.** If , , and are not all 0, show that the equation

represents a plane and is a normal vector to the plane.

*Hint: Suppose * and rewrite the equation in the form

**82.** Give a geometric description of each family of planes.

(a) (b)

(c)

*L*1 共2, 0, ⫺1兲

*L*2 共1, ⫺1, 1兲 共4, 1, 3兲

*L*1 *L*2

*L*1 共1, 2, 6兲 共2, 4, 8兲

*L*2 ^{1} ^{2}

^{1} *x⫺ y ⫹ 2z ⫹ 1 苷 0* ^{2}

共3, 2, ⫺1兲 共0, 0, 1兲 共1, 2, 1兲
*L*1 *L*2

*a b* *c*

*ax⫹ by ⫹ cz ⫹ d 苷 0* *具a, b, c典*

*a*苷 0

*a*

## 冉

^{x}^{⫹}

^{d}^{a}## 冊

*⫹ b共y ⫺ 0兲 ⫹ c共z ⫺ 0兲 苷 0*

*x⫹ y ⫹ z 苷 c* *x⫹ y ⫹ cz 苷 1*

*y cos ⫹ z sin 苷 1*

*x苷 1 ⫹ 2s*
*z苷 2t*

*y苷 1 ⫹ 6t*
*x苷 1 ⫹ t*

*z苷 ⫺2 ⫹ 6s*
*y苷 5 ⫹ 15s*

**L A B O R AT O RY P R O J E C T** **PUTTING 3D IN PERSPECTIVE**

Computer graphics programmers face the same challenge as the great painters of the past: how
to represent a three-dimensional scene as a flat image on a two-dimensional plane (a screen or a
canvas). To create the illusion of perspective, in which closer objects appear larger than those
farther away, three-dimensional objects in the computer’s memory are projected onto a rect-
angular screen window from a viewpoint where the eye, or camera, is located. The viewing
volume––the portion of space that will be visible––is the region contained by the four planes that
pass through the viewpoint and an edge of the screen window. If objects in the scene extend
beyond these four planes, they must be truncated before pixel data are sent to the screen. These
*planes are therefore called clipping planes.*

**1.** Suppose the screen is represented by a rectangle in the -plane with vertices

and , and the camera is placed at . A line in the scene passes through the points and . At what points should be clipped by the clipping planes?

**2.** If the clipped line segment is projected on the screen window, identify the resulting line
segment.

**3.** Use parametric equations to plot the edges of the screen window, the clipped line segment,
and its projection on the screen window. Then add sight lines connecting the viewpoint to
each end of the clipped segments to verify that the projection is correct.

**4.** A rectangle with vertices , , , and

is added to the scene. The line intersects this rectangle. To make the rect-
*angle appear opaque, a programmer can use hidden line rendering, which removes portions*
of objects that are behind other objects. Identify the portion of that should be removed.

*yz* 共0, ⫾400, 0兲

共0, ⫾400, 600兲 共1000, 0, 0兲 *L*

共230, ⫺285, 102兲 共860, 105, 264兲 *L*

共621, ⫺147, 206兲 共563, 31, 242兲 共657, ⫺111, 86兲

共599, 67, 122兲 *L*

*L*
98845_ch12_ptg01_hr_848-857.qk_98845_ch12_ptg01_hr_848-857 8/18/11 3:28 PM Page 850