8 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE
Example 1.3. Calculate the following:
(a) Th e magnitude of the vector
−−→
PQ in R
2
with P =(−1,2) and Q =(5, 5).
Solution: By formula (1.2),
°
°
−−→
PQ
°
°
=
p
(5 −(−1))
2
+(5 −2)
2
=
p
36 +9 =
p
45 =3
p
5.
(b) Th e magnitude of the vector v =(8, 3) in R
2
.
Solution: By formula (1.3), kvk=
p
8
2
+3
2
=
p
73.
(c) T he distance between the points P =(2, −1,4) and Q =(4, 2,−3) in R
2
.
Solution: By formula (1.4), the distance d =
p
(4 −2)
2
+(2 −(−1))
2
+(−3 −4)
2
=
p
4 +9 +49 =
p
62.
(d) The magnitude of the vector v =(5,8,−2) in R
3
.
Solution: By formula (1.5), kvk=
p
5
2
+8
2
+(−2)
2
=
p
25 +64 +4 =
p
93.
Exercises
A
1. Ca lculate the magnitudes of the following vectors:
(a) v =(2,−1) (b) v =(2,−1,0) (c) v =(3,2,−2) (d) v =(0,0,1) (e) v =(6,4,−4)
2. For the points P =(1,−1,1), Q =(2,−2,2), R =(2,0,1), S =(3, −1,2), does
−−→
PQ =
−−→
RS?
3. For the points P =(0,0,0), Q =(1,3,2), R =(1,0,1), S =(2, 3,4), does
−−→
PQ =
−−→
RS?
B
4. Le t v =(1, 0,0) and w =(a, 0,0) be vectors in R
3
. Show that kwk=|a|kvk.
5. Le t v =(a, b, c ) and w =(3a,3b,3c) be vectors in R
3
. Show that kwk=3kvk.
C
x
y
z
0
P(x
1
, y
1
, z
1
)
Q(x
2
, y
2
, z
2
)
R(x
2
, y
2
, z
1
)
S(x
1
, y
1
,0)
T(x
2
, y
2
,0)
U(x
2
, y
1
,0)
Figure 1.1.9
6. Th ough we will see a simple proof of Theorem 1.1
in the next section, it is possible to prove it using
methods similar to those in the proof of Theorem
1.2. Prove the special case of Theorem 1.1 where the
points P =(x
1
, y
1
, z
1
) and Q =(x
2
, y
2
, z
2
) satisfy the fol-
lowing conditions:
x
2
> x
1
>0, y
2
> y
1
>0, and z
2
> z
1
>0.
(Hint: Think of Case 4 in the proof of Theorem 1.2,
and consider Figure 1.1.9.)
剩余221页未读，继续阅读
patrickpdx
- 粉丝: 391
- 资源: 8
我的内容管理
展开
最新资源
- 多模态联合稀疏表示在视频目标跟踪中的应用
- Kubernetes资源管控与Gardener开源软件实践解析
- MPI集群监控与负载平衡策略
- 自动化PHP安全漏洞检测：静态代码分析与数据流方法
- 青苔数据CEO程永：技术生态与阿里云开放创新
- 制造业转型： HyperX引领企业上云策略
- 赵维五分享：航空工业电子采购上云实战与运维策略
- 单片机控制的LED点阵显示屏设计及其实现
- 驻云科技李俊涛：AI驱动的云上服务新趋势与挑战
- 6LoWPAN物联网边界路由器：设计与实现
- 猩便利工程师仲小玉：Terraform云资源管理最佳实践与团队协作
- 类差分度改进的互信息特征选择提升文本分类性能
- VERITAS与阿里云合作的混合云转型与数据保护方案
- 云制造中的生产线仿真模型设计与虚拟化研究
- 汪洋在PostgresChina2018分享：高可用 PostgreSQL 工具与架构设计
- 2018 PostgresChina大会：阿里云时空引擎Ganos在PostgreSQL中的创新应用与多模型存储
资源上传下载、课程学习等过程中有任何疑问或建议，欢迎提出宝贵意见哦~我们会及时处理！
点击此处反馈
